Finite and Infinite Rotation sequences and Beyond · 2019-02-20 · Finite and Infinite Rotation Sequences and Beyond Dissertation im Fachbereich Mathematik zur Erlangung des Grades

Finite and Infinite Rotation Sequencesand Beyond

Dissertation

im Fachbereich Mathematik zur

Erlangung des GradesDr. rer. nat.

vonArne Mosbach

Vorgelegt am26.11.2018

Version:OR 204 Zero

ii

Gutachter: Prof. Dr. Marc Keßebohmer (Universitat Bremen)Prof. Dr. Daniel Lenz (Universitat Jena)

iii

Abstract

The encoding of orbits attained from rigid rotations are in-vestigated from different perspectives. In the first part of thethesis regularity conditions for irrational rotations will bestudied in terms of their continued fraction expansions and acategorisation is achieved for continued fraction expansionswhich do not grow too fast. The second part focuses on thespectral properties of β-transformations for β ≤

√2. Here

an explicit representation for the Bochner transform of au-tocorrelations stemming from Dirac combs derived from β-transformations is achieved, which consists of a Lebesgue-absolutely continuous part and a discrete part. The lastpart focuses on vague limits of these autocorrelations whereβ→ 1. Here a link to subshifts derived from rigid rotationswill be established. The Bochner transform of these vaguelimits can be given explicitly in some cases and is shown tobe either discrete, non-discrete singular to Lebesgue, or amixture of both.

Ubersicht

Die Arbeit befasst sich mit der Kodierung von Rotati-onsabbildungen. Hierzu wird im ersten Teil die Kodie-rung uber Kettenbruche irrationaler Zahlen eingefuhrt unddie Regularitatseigenschaften der daraus abgeleiteten Sub-shifts untersucht. Erreicht wird eine Klassifizierung allerKettenbruche deren Eintrage nicht extrem schnell wach-sen. Der nachste Teil befasst sich mit den Spektraleigen-schaften von β-Transformationen fur β ≤

√2. Dazu wird

die Bochnertransformierte zur Autokorrelation eines re-turn time combs zu einer β-Transformation gebildet undeine explizite Darstellung ebendieser gegeben. Diese be-sitzt Lebesgue-absolut stetigen und einen diskreten Teil. Alsnachstes wird fur β → 1 eine Verbindung zu der eingangsgegebenen Kodierung von Rotationsabbildungen aufgezeigtund die Bochnertransformierte kann in einigen Fallen auchdort explizit bestimmt werden. In den Fallen ist sie ent-weder diskret, singular zum Lebesguemaß oder eine Misch-ung aus beidem.

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Contents

1 Introduction 11.1 Exposition and main results . . . . . . . . . . . . . . . . . . . . . 31.2 Outline of chapters . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Continued fractions and symbolic representation 212.1 Continued fraction expansion . . . . . . . . . . . . . . . . . . . . 212.2 Symbolic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Substitutions of rotation type . . . . . . . . . . . . . . . . . . . . 31

3 Holder regularity for irrational numbers and their subshifts 393.1 Bounds on continued fraction expansion . . . . . . . . . . . . . . 39

3.1.1 Sturmian subshifts of slope ξ . . . . . . . . . . . . . . . . 403.2 Right special factors in Sturmian subshifts . . . . . . . . . . . . . 413.3 Spectral metrics on Sturmian subshifts . . . . . . . . . . . . . . . 49

3.3.1 Subsequences of ψz approximands . . . . . . . . . . . . . 523.3.2 Estimates on ψw . . . . . . . . . . . . . . . . . . . . . . . 633.3.3 Holder regularity and continued fraction expansion . . . . 65

3.4 Hausdorff dimension of Θα . . . . . . . . . . . . . . . . . . . . . 70

4 Measure theory 734.1 Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Non-negative measures . . . . . . . . . . . . . . . . . . . . . . . 744.3 Complex-valued measures . . . . . . . . . . . . . . . . . . . . . 754.4 Decomposition of a measure . . . . . . . . . . . . . . . . . . . . 76

5 Fourier transformation and Bochner’s theorem 795.1 Fourier transform of functions . . . . . . . . . . . . . . . . . . . 795.2 Bochner’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.1 The Integers . . . . . . . . . . . . . . . . . . . . . . . . 84

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vi CONTENTS

6 Dynamical systems 876.1 Operator theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.1.1 Spectrum of an operator . . . . . . . . . . . . . . . . . . 886.1.2 Isometric isomorphisms or unitary operators . . . . . . . 89

6.2 Spectral measures . . . . . . . . . . . . . . . . . . . . . . . . . . 896.3 Return time combs . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.3.1 Rotations on the unit circle . . . . . . . . . . . . . . . . . 946.4 Spectrum for isometric isomorphisms . . . . . . . . . . . . . . . 1016.5 Operator on dynamical systems . . . . . . . . . . . . . . . . . . . 102

6.5.1 Spectrum for weakly mixing systems . . . . . . . . . . . 1086.6 Substitution dynamical systems . . . . . . . . . . . . . . . . . . . 111

7 Quasicrystals 1177.1 Cut and Project Schemes . . . . . . . . . . . . . . . . . . . . . . 118

7.1.1 Rotation as Cut and Project Scheme . . . . . . . . . . . . 119

8 β-transformations 1218.1 Symbolic space for β-transformations . . . . . . . . . . . . . . . 1218.2 Categorisation of β-transformations . . . . . . . . . . . . . . . . 1238.3 The Parry measure . . . . . . . . . . . . . . . . . . . . . . . . . 1278.4 Spectral properties . . . . . . . . . . . . . . . . . . . . . . . . . 1318.5 Substitutions from β-transformations . . . . . . . . . . . . . . . . 142

8.5.1 Thue-Morse substitution . . . . . . . . . . . . . . . . . . 1548.5.2 Variation of the Thue-Morse case . . . . . . . . . . . . . 157

8.6 Convergence of β-transformations . . . . . . . . . . . . . . . . . 166

A Dynamical systems 169A.1 Conjugacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

B Functional analysis 171B.1 Separability of Lp spaces . . . . . . . . . . . . . . . . . . . . . . 171B.2 Topologies and metrics . . . . . . . . . . . . . . . . . . . . . . . 171

B.2.1 Normed spaces . . . . . . . . . . . . . . . . . . . . . . . 172B.2.2 Vague topology . . . . . . . . . . . . . . . . . . . . . . . 172

B.3 The space C′c . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

C Fourier methods 175C.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

C.1.1 Dual group and characters . . . . . . . . . . . . . . . . . 177C.2 Fourier transformation of functionals . . . . . . . . . . . . . . . . 179C.3 Some properties of Fourier transformation . . . . . . . . . . . . . 181

Chapter 1

Introduction

One way to describe internal structures of a given (physical or mathematical) ob-ject or process is by analysing the occurrences of patterns on a certain scale, suchas time, length, etc. One of the simplest cases is a purely periodic structure. Itcan be defined in a rigorous way for many purposes in mathematics. A profoundexample, studied in this thesis, of such a behaviour is given by a rotation on theunit interval [0, 1) modulo 1 with a rational number α = p/q ∈ Q between 0 and1, which can be written as a map T (x) = x + α mod 1. The orbit of 0 is givenby all k · p/q mod 1, where k is a natural number and is periodic, as for k = q acalculation shows q · p/q = p = 0 mod 1.

Beyond periodicity one can still look for structure in an object, which is thenreferred to as aperiodic. For aperiodicity it turns out to be a lot more difficult to setup a mathematical concept. A leading example is given if we choose an irrationalα between 0 and 1 and look at the orbits of 0 for T (x) = x + α mod 1. More so,for a rational and an irrational number their orbits only have 0 in common andare different everywhere else. Regardless, if an irrational number α is approxi-mated by rational numbers, their orbits almost match for many iterates, but aftera large number of iterations it is hard to predict if they are close to each otheror far away. This pattern repeats every time when two points of their orbits arevery close to each other and this regularity in “how close” the rational approxi-mates for an irrational number are is reflected in the continued fraction expansionof these numbers. An explanation can be given by the Diophantine approxima-tion of numbers of which the ones given by the continued fraction expansion areremarkable. The continued fraction expansion of a number α can be obtainedwith the euclidean algorithm by noting down every remainder starting with 1/α.With that the continued fraction expansion of an irrational α presents rational ap-proximations of α by putting the euclidean algorithm at some point to a hold. Ahuge difference between two consecutive entries in the continued fraction impliesthat the numbers are almost the same, similar as to the difference between 0 and

1

2 CHAPTER 1. INTRODUCTION

1/1013. This establishes a link between bounds on the growth of the continuedfraction entries and aperiodicity.

Another way of looking at aperiodicity is by introducing a 2-letter coding(e.g. 0 and 1) for each number α ∈ [0, 1) related to its orbit by noting down a1 if αn − ⌊αn⌋ > 1 − α and 0 otherwise for any natural number n. In this waythe coding for a rational given via a truncated continued fraction expansion of anirrational appears at infinite many places in the coding of the irrational, so onecan say it returns infinitely often. Roughly speaking one wants describe structureswhich exhibit repeating patterns that are the same, but not on a regular scale.A quantitative research of these occurrences in connection to the size of theircontinued fraction entries is done in Chapter 3 by introducing tools to describethe regularity of these codings.

Another approach to aperiodic behaviour is given by the autocorrelation and itsFourier transform. The autocorrelation can be seen as a comparison of an objectwith a shifted copy of itself. They are often considered in physics e.g. signalprocessing and quasicrystals, where in the latter one the study of aperiodicity isthe driving factor. This is also implied by its name, as crystals are modelled asperiodic structures, while a quasicrystal is not periodic, but still a lot of structurecan be seen in it. So what would happen if a periodic object is approximated byaperiodic objects and vice versa. For rotations that is α is approximated by somesequence and this will carry over to a sequence of autocorrelations converging tothe autocorrelation of ω, which is covered in Section 6.3.1. There, the definingfeature is the map T (x) = x+α mod 1 and one could ask what happens if a slightlydifferent one is chosen. Introducing another parameter 1 < β < 2 we set T (x) =βx + α mod 1, which are called β-transformations. One distinct feature is that themaps now have a discontinuity and are not anymore bijective (invertible) as theirslope is larger than 1. These seemingly small changes have huge consequencesand get one closer to chaotic behaviour. Indeed β-transformations can be linkedto Lorenz maps, but regardless an autocorrelation can be constructed with them.Chaos is related to entropy, which measures the disarray in an object. One canthink of periodicity (crystals) as order, chaos (amorphous) as total disorder andaperiodicity (quasicrystals) as something inbetween which is much closer to order.One can then ask if the autocorrelations of β-transformations exhibit more chaoticor aperiodic features and, if β tends to one, do the autocorrelations then convergewith the ones derived from rotation and does a limit even exist. This will be lookedupon in Chapter 8.

1.1. EXPOSITION AND MAIN RESULTS 3

1.1 Exposition and main resultsAny ξ ∈ [0, 1) induces the map T (x) ≔ {x + ξ} ≔ x + ξ mod 1 and hence thedynamical system ([0, 1),B,T ), where B denotes the Borel-σ-algebra which isalso given in the Nomenclature. If the orbits are periodic, the number ξ is rationaland there exists an n ∈ N such that

ξ = pn/qn = [0; a1 + 1, . . . , an] =1

(a1 + 1) + 1a2+

1

...+ 1an

for some a1 ≥ 0, ai ≥ 1 for all i ∈ {2, . . . , n} and numbers pn, qn ∈ N withgcd(pn, qn) = 1. The number ξ can be uniquely identified with the finite 0-1-sequence κ = (1[0,ξ) ◦ T j(ξ))qn−1

j=0 induced by its orbit starting at 0. Another repre-sentation of the rotation-sequence κ as a word can be obtained by exploiting thecontinued fraction expansion of ξ via substitutions (there is an abuse of notationby writing e.g. 01 instead of (0, 1))

τ :

⎧⎪⎪⎨⎪⎪⎩0 ↦→ 01 ↦→ 10

, ρ :

⎧⎪⎪⎨⎪⎪⎩0 ↦→ 011 ↦→ 1

,

such that κ = τa1ρa2 · · · τan−2ρan−1(10) if n is even and κ = τa1ρa2 · · · ρan−2τan−1(10) ifn is odd, see Theorem 2.3.8. This relation extends to irrational ξ by taking n to∞,where ξ is approximated by its continued fraction expansion and leads to Sturmiansubshifts of slope ξ, see Definition 3.1.3. This representation has for example beenstudied in [62, 63, 29, 70], where it is also shown that a Sturmian subshift of slopeξ is minimal and ergodic. In [63, 29, 70, 47, 46] regularity conditions such aslinearly repetitive, repulsive and power free are introduced, see Remarks 2.2.13and 2.2.17 for a definition. The authors have shown that these regularity condi-tions are equivalent for Sturmian subshifts of slope ξ and furthermore equivalent tothe continued fraction entries of ξ being bounded. Here α-regularity conditions,α-repetitive, α-repulsive and α-finite are given in Definitions 2.2.12 and 2.2.14,which have been studied in [35] joint with Groger, Keßebohmer, Samuel and Stef-fens. For Sturmian subshifts of slope ξ and α = 1 it is shown in Remarks 2.2.13and 2.2.17 that the α-regularity conditions coincide with the former regularityconditions. Here for any α ≥ 1 we set Aα(ξ) ≔ lim supn→∞ anq1−α

n−1 and define

Θα≔ {ξ ∈ [0, 1]\Q : 0 < Aα(ξ)}, Θα ≔ {ξ ∈ [0, 1]\Q : Aα(ξ) < ∞}

and Θα ≔ Θα ∩ Θα. With that, for α > 1, these results are complemented andextended by Theorem 3.2.6 which says

Theorem. For α > 1 and ξ ∈ [0, 1] irrational, the following are equivalent.


1. The Sturmian subshift of slope ξ is α-repetitive.

2. The Sturmian subshift of slope ξ is α-repulsive.

3. The Sturmian subshift of slope ξ is α-finite.

4. For the Sturmian subshift of slope ξ we have ξ ∈ Θα.

The canonical choice for a metric on a subshift is given by d(v,w) = 2−|v∧w|,where v,w are distinct elements of the subshift and v∧w denotes the longest prefixthey have in common. In the case that v and w are equal it is set to be zero. Thesame topology is generated by the metrics dt = |v ∧ w|−t, where t > 0, which areare considered in this work. Their slower (polynomial) growthrate will be used tostudy aperiodic behaviour of subshifts. On Sturmian subshifts of slope ξ a spectralmetric dξ,t can be defined, see Definition 3.3.2, by putting additional weight withrespect to (n−t)n∈N on every right special factor (see Section 2.2 for a definition).This is done via a function bn(z), which is equal to 1 if z|n is a right special wordof the subshift and 0 otherwise. The spectral metric is then given by

dξ,t(v,w) ≔ |v ∨ w|−t +∑

n>|v∨w|

bn(v)n−t +∑

n>|v∨w|

bn(w)n−t,

for all v,w ∈ X. This is a modification of spectral triples introduced in [11] andfurther generalised and extended in [23, 36, 37, 46]. The definition for a spectraltriple and the spectral metric considered in this work is given in [47, 46] and willnot be pursued further. In Section 3.3 both structures dξ,t and dt are compared toeach other with a Holder regularity condition, that is for any r > 0 given by

ψw(r) ≔ lim supv−→

dtw

dξ,t(w, v)dt(w, v)r and ψ(r) ≔ sup{ψw(r) : w ∈ X},

where w is an element of the Sturmian subshift, see (3.10). Further we define inDefinition 3.3.5 the following notions

1. The metric dξ,t is sequentially r-Holder regular to dt if ψ(r) < ∞.

2. The metric dξ,t is sequentially r-Holder regular to dt if ψ(r) > 0.

3. The metric dξ,t is sequentially r-Holder regular to dt if 0 < ψ(r) < ∞.

It turns out that for α > 1 and a Sturmian subshift of slope ξ ∈ Θα, the spectral

metric dξ,t is not a metric for any t ∈ (0, 1 − 1/α], however for t ∈ (1 − 1/α,∞),the spectral metric dξ,t is a metric, see Proposition 3.3.3. This is emphasised bythe following definition of the function ϱα(t) which is 0 if t ≤ 1 − 1/α, equal to1 − (α − 1)/(αt) if 1 − 1/α < t < 1 and 1/α if t ≥ 1. Therefore it will generally beassumed that t > 1 − 1/α from here on, such as in Theorem 3.3.15 which statesthe following.


Theorem. Let X be a Sturmian subshift of slope ξ, let α > 1 be given and fixt ∈ (1 − 1/α, 1).

1. The metric dξ,t is sequentially ϱα(t)-Holder regular to the metric dt if andonly if ξ ∈ Θα.

2. The metric dξ,t is sequentially ϱα(t)-Holder regular to the metric dt if andonly if ξ ∈ Θ

α.

Hence, dξ,t is sequentially ϱα(t)-Holder regular to dt if and only if ξ ∈ Θα.

A connection to regularity conditions on a Sturmian subshift of slope ξ ∈Θα is established via Theorem 3.2.6. As we have just seen for any irrationalnumber ξ in Θ

α,respectively Θα, for some α > 1 we only have sequentially ϱα(t)-

Holder regular, respectively sequentially ϱα(t)-Holder regular, for some t < 1. TheDefinition 3.3.6 of critical sequential Holder regularity might be satisfied t ≥ 1.That is for any r > 0 given as follows.

1. The metric dξ,t is critically sequentially r-Holder regular to dt if ψ(r−ϵ) = 0,for all 0 < ϵ < r.

2. The metric dξ,t is critically sequentially r-Holder regular to dt if ψ(r + ϵ) =∞, for all ϵ > 0.

3. The metric dξ,t is critically sequentially r-Holder regular to dt if dξ,t is criti-cally sequentially r- and r-Holder regular to dt.

Critically sequentially Holder regular is a weaker notion than sequentially Holderregular. Thus Theorem 3.3.15 still holds in one direction and for the critical choicet = 1, by Theorem 3.3.16, when sequential ϱα(t)-Holder regularity, respectivelysequential ϱα(t)-Holder regularity, cannot be implied for some ξ in Θ

α, respec-

tively Θα, then critical sequential Holder regularity is still satisfied with respect toϱα for t ≥ 1. This is given in detail in Theorem 3.3.16 stated in the following.

Theorem. Let X be a Sturmian subshift of slope ξ and let α > 1 be real.

1. For t = 1, we have the following.

(a) If dξ,t is sequentially 1/α-Holder regular to dt, then ξ ∈ Θα.

(b) If ξ ∈ Θα, then dξ,t is critically sequentially 1/α-Holder regular to dt.

(c) If ξ ∈ Θα, then dξ,t is critically sequentially 1/α-Holder regular to dt.

2. For t > 1, we have the following.


(a) If ξ ∈ Θα, then dξ,t is sequentially 1/α-Holder regular to dt.

(b) If ξ ∈ Θα, then dξ,t is sequentially 1/α-Holder regular to dt.

3. (a) If t ∈ (1, α/(α− 1)) and if dξ,t is sequentially 1/α-Holder regular to dt,then ξ ∈ Θα.

(b) If t ≥ α/(α − 1), then dξ,t is 1/α-Holder continuous with respect to dt.

By using results from [52, 22] there is even an estimate on the Hausdorff di-mension for Θα,Θα and Θ

αrespectively, where α > 1, see Theorem 3.4.3, which

is also given in [83] and reads as follows.

Theorem. For α > 1 we have that dimH (Θα) = dimH (Θα) = 2/(α + 1) and

m(Θα) = 1, where dimH denotes the Hausdorff dimension and m the Lebesguemeasure on R.

In the second part, we study the regularity of rigid rotations, β-transformationsand their associated subshifts in terms of their spectral properties. A primer of thisis [50], jointly with Keßebohmer, Samuel and Steffens, which is to appear in theJournal of Statistical Physics and the results obtained there will be applied andgeneralised within this thesis in the context of β-transformations. Let us con-sider an erdogic dynamical system (X,B,T, ν), where T is ν-invariant and a ν-measurable bounded function f : X → C and y ∈ X. These allow us to define aDirac comb on Z by

ηy ≔

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩∑n∈N

f ◦ T n(y) δn, if T is non-invertible∑z∈Z

f ◦ T z(y) δz, if T is invertible

which is called an f -weighted return time comb with respect to T and referencepoint y. In the theory of quasicrystals there is a huge interest in the autocorrelationof such structures, which can be seen as a smoothing operator of a function ina symmetric way. It can be defined in many different ways and has even beengeneralised up to locally compact abelian groups, see [8, 9, 65, 57, 73, 75] forsome of the works constructing autocorrelations or generalisations of them. Here,whenever it exists, we denote the autocorrelation of a weighted return time combηy by γy or ηy ~ ˜ηy, which is then defined as

v-limn→∞

ηy |[−n,n] ∗ ˜ηy |[−n,n]

n + 1, if T is non-invertible,

v-limn→∞

ηy |[−n,n] ∗ ˜ηy |[−n,n]

2n + 1, if T is invertible,


where ˜ηy( f ) = ⟨ηy, f (−·)⟩ for all f ∈ Cc(Z). With a result of [80] and Birkhoff’sergodic theorem the existence of a vague limit can be guaranteed a.s. and thus γy

exists, which is a widely-used concept, see e.g. [9, 58].

Theorem. For an f -weighted return time comb ηy with respect to T and withreference point y, the autocorrelation γy exists for ν-almost every y and equals

γηy =∑z∈Z

Ξ(T, ν)(z) δz,

where

Ξ(T, ν)(z) ≔

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫f ◦ T |z| · f dν, if T is non-invertible and z ≥ 0,

∫f ◦ T |z| · f dν, if T is non-invertible and z < 0,

∫f ◦ T−z · f dν, if T is invertible.

The difference to the works mentioned above is that T is not required to be in-vertible and hence the dynamical system cannot be induced by a group action. Thedefinition of Ξ(T, ν) implies that it is a positive definite function on Z and thus, byTheorem 5.2.2, has a Bochner transform. Bochner’s theorem states that there ex-ists a unique non-negative finite measure for every positive definite function on Zand the transformation is a homeomorphism, see [13, 79]. This homeomorphismcan be extended to SCP(Z), the span of positive definite functions on Z, and thusthe Bochner transform is a complex-valued measure on [0, 1) due to the Jordan-decomposition of a measure. With that and some additional work, Corollary 6.3.5states that for an f -weighted return time comb ηy and a g-weighted return timecomb η′y, which are given with respect to an ergodic system (X,B,T, ν), the mea-sure ηy ~ ˜η′y exists for ν-almost every y ∈ X and is given by

ηy ~ ˜η′y(z) =

⎧⎪⎪⎨⎪⎪⎩∫

f ◦ T |z| · g dν, z ≥ 0∫f · g ◦ T |z| dν, z < 0

and the Bochner transform of ηy ~ ˜η′y exists. This establishes a nice relation tospectral measures, which are for example defined in [70, 71], see (6.1) and (6.4).Bochner’s theorem has experienced many generalisations and alternative formu-lations, see [3, 13, 59], which envelope the statements made here. As the Bochnertransform is a continuous operator and SCP(Z) is closed, see e.g. [13, 79], one canalso study the vague limit of autocorrelations and transfer the results to the vaguelimit of the Bochner transforms. A first example of that is done for rigid rotationson the unit circle which is given in Theorem 6.3.10 and stated in the following.


Theorem. Let f : [0, 1) → R≥0 be Riemann integrable and let α ∈ R+. Fix asequence ( fi)i∈N of non-negative Riemann integrable functions that converges uni-formly to f on [0, 1), and fix a sequence (αi)i∈N in R+ with limi→∞ αi = α andαi ≠ α for all i ∈ N. Let (yi)i∈N denote a sequence of reference points in [0, 1) and,for i ∈ N, let µyi denote the fi-weighted return time comb with respect to Tαi andwith reference point yi. The sequence of autocorrelations (γµyi

)i∈N attains a vaguelimit γ given by

γ =∑z∈Z

Ξ(Tα,m|[0,1))(z) δz

and

ˆγ =∑m∈Z

ˆΞ(Tα,m|[0,1))(m) δ{mα}.

In particular ˆΞ(Tα,m|[0,1))(m) = |ˆf |2(m) for all m ∈ N.

With this at hand, for a sequence (αi)i∈N with limi→∞ αi = α and αi ≠ α forall i ∈ N, where α = p/q ∈ Q, one obtains a vague limit given by Ξ(Tα,m|[0,1)),which is in general not equal to Ξ(Tα, µq,y), where µq,y is an ergodic measure of therational rotation by α, see Section 6.3.1 for a precise definition. This complementsa result of Beckus and Pogorzelski in [10], who showed convergence in the caseof unique ergodicity for Cut and Project Schemes, which are in this case given by(R, [0, 1), {(n, {nα}) : n ∈ Z}) for any α ∈ R+.

For weakly mixing dynamical systems ([0, 1],B,T, µ) it is known that theKoopman operator of T on L2

µ([0, 1]) has only the eigenvalue 1 and its eigenfunc-tions are constant, see [30]. In [40] a Perron-Frobenius operator is defined if Tadmits inverse branches, however within this work, T is a piecewise differentiabletransformation with its derivative being of bounded variation which is given inDefinition 6.5.4. The operator P only has the eigenvalue 1 if the dynamical systemis weakly mixing, see [40, 48]. In particular, P( f ) = h

∫f dm + Ψ( f ) for every f

of bounded variation, where h is a non-negative function of bounded variation andΨ is an operator with ∥Ψn∥ ≤ Csn for some C > 0, s ∈ (0, 1), see Theorem 6.5.7,where the operator norm is taken with respect to the space of functions of boundedvariation, which is denoted by BV. This can be combined with Bochner’s theoremto obtain Theorem 6.5.13 given in the following.

Theorem. Let ([0, 1],B,T, µ) be weakly mixing, where µ = hm, and T admitsinverse branches. Further let f1, f2 ∈ BV be real valued. Further denote by ηi

the fi-weighted return time comb with respect to T and with reference point y fori ∈ {1, 2}. Then for µ-almost every y the spectral return measure ˆη1 ~ ˜η2 of η1 ~ ˜η2


is given by

ˆη1 ~ ˜η2 =

∫f1 dµ

∫f2 dµ δ0 + gm.

Here g(x) ≔∑

z∈Z cz e2πixz, where cz ≔∫

f1 · Ψ|z|( f2h) dm, for z > 0, cz ≔

∫f2 ·

Ψ|z|( f1h) dm, for z < 0 and c0 ≔∫

f1 f2 dµ−∫

f1 dµ∫

f2 dµ is an analytic function.

D1,2

D((1,2);(1,3))D((1,2);(5,7))

D((1,2);(1,3);(1,5))

D((1,2);(1,3);(3,5))

Figure 1.1: An example of Dℓ for ℓ equal to ((1, 2)), ((1, 2); (1, 3)). ((1, 2); (5, 7)),((1, 2); (1, 3); (1, 5)) and ((1, 2); (1, 3); (3, 5)) respectively.

For (intermediate) β-transformations T (x) = {βx + α}, where (β, α) ∈ ∆ ≔{(β, α) ∈ R2

+ : β > 1, 0 ≤ α ≤ 2 − β} one has that h is the Parry density givenin [67, 69], see (8.2). The dynamical system ([0, 1),B,T, µ), where µ = hm,is not weakly mixing if and only if there exist k, n ∈ N, k < n, gcd(k, n) = 1such that (β, α) ∈ Dk,n ⊆ ∆, see Theorem 8.4.1 and Definition 8.2.8 for a defini-tion of the areas Dk,n. These areas have been determined by Palmer in [66] andwere further studied in [33]. Palmer also showed for any T (x) = {βx + α}, where(β, α) ∈ Dk,n, we have that T n restricted on an interval in [0, 1] is measure theoret-ically isomorphic to another β-transformation up to a constant, see Lemma 8.3.3.This can be used in a reversed direction with techniques from [33] to obtain abijective mapping from ∆ to Dk,n, see Lemma 8.3.1. A consecutive applicationof that mapping for a finite sequence ℓ ≔

((ki, ni)

)mi=0 for m ∈ N, where ki < ni

with gcd(ki, ni) = 1, yields a set which will be called Dℓ ⊆ ∆, see Figure 1.1 andDefinition 8.4.7 for a precise definition. For any (β, α) ∈ Dℓ, T (x) = {βx + α}, onehas that by construction T q, for q ≔

∏mj=0 n j, restricted on some interval in [0, 1)

is weakly mixing and measure theoretically isomorphic to a β-transformation upto a constant. There is a general principle behind it, as for any dynamical systemthat satisfies the conditions for the theorem of Ionescu-Tulcea and Marinescu todefine a Perron Frobenius operator as mentioned above, Hofbauer and Keller men-tioned in [40, 48] that the system can be split up into finitely many components


0 20 40 60 80 100

Figure 1.2: For (β, α) ≈ (1.0385, 0.1799) ∈ D((1,5);(1,3)) and f = 1[c,1), wherec = (1 − α)/β the first 100 entries of the corresponding autocorrelation γ areshown.

which are weakly mixing with respect to some power of the transformation. Forβ-transformations this can be calculated explicitly and is given in Theorem 8.4.10,of which in the following a simpler version that foregoes an explicit descriptionof every variable is presented. Note that any complex-valued function η on Z canbe split up with respect to q into functions by defining for any 0 ≤ r ≤ q − 1 thefunction η(q,r)(z) by η(zq) if zq ∈ Z such that z = zq + r and 0 otherwise.

Theorem. Let ℓ =((ki, ni)

)mi=0, where gcd(ki, ni) = 1 for all 0 ≤ i ≤ m. Denote by

ηy an f1-weighted return time comb and by η′y an f2-weighted return time comb,both with respect to T0 : x ↦→ {β0x + α0} and reference point y, where (β0, α0) ∈Dℓ. Then there exists (βq

0, αm+1) ∈ ∆ and weighted return time combs νi,r, ν′i withrespect to Tm+1(x) = {βq

0x + αm+1} for all 0 ≤ r ≤ q − 1 and i ∈✕m

j=0{1, . . . , n j}

such that for hm|[0,1)-a.e. y

ηy ~ ˜η′y = 1q

∑i∈

✕mj=0{1,...,n j}

q−1∑r=0

(νi,r ~ ˜ν′i)(q,r)

,

where q ≔∏m

j=0 n j and h denotes the Parry density to T0.

For each fixed i ∈✕m

j=0{1, . . . , n j} the measures on the right hand side ofthe theorem correspond to a measure theoretical dynamical system and r onlychanges the density. All of them are measure theoretically isomorphic to the dy-namical system ([0, 1),B,Tm+1, µm+1), where Tm+1(x) = β

q0x + αm+1 and µm+1 is


its Parry measure. With this representation the spectral return measure for any β-transformation can be calculated, which is referred to in Theorem 8.4.11 and Re-

mark 8.4.12, and presented here in a simpler form.

Theorem. Let ηy � η′y be given as in the previous theorem, where additionallyf1, f2 ∈ BV are real-valued and further assume (βm+1, αm+1) � Dk,n for all k, n ∈ Nwith gcd(k, n) = 1. Then its Bochner transform is given by

(ηy � η′y)∧ =

1

q2

∑i∈�m

j=0{1,...,n j}

q−1∑r=0

Ci,re−r δ 1qZq+ e−r(gi,rm|[0,1/q) ∗ δ 1

qZq),

where Ci,r ∈ R is a constant and gi,r is a complex analytic function for all 0 ≤ r ≤q − 1 and i ∈

�mj=0{1, . . . , nj}.

Figure 1.3: For (β, α) ≈ (1.0385, 0.1799) ∈ D((1,5);(1,3)) the spectral return measure

of the autocorrelation γ (see Figure 1.2) is given. Some of the peaks have been

cut off in the picture and the one at 0 is omitted. All peaks are given by multiples

of k/(3 · 5) for k ∈ {0, 1, . . . , 14}.

In the next part we consider return time combs with weight functions 1[0,γ),

where γ � (1 − α)/β for a β-transformation T will be called T -discontinuity


and is chosen such that T (γ) = 0. This will be in context to certain sequences((βi, αi))i∈N, where (βi, αi) → (1, α) for some α ∈ (0, 1), which induce a sequenceof β-transformations and hence a sequence of the mentioned weighted return timecombs. The vague limit of the autocorrelations and spectral return measures ofthese return time combs is then the object of our interest. As βi → 1, the β-transformations approximate a rigid rotation, which is linked to the substitutionsτ and ρ introduced at the beginning of the introduction. One can associate asubstitution σ to any rational k/n for a set Dk,n in the following way. Set

Q ≔{τa1ρa2τa3 . . . τa j−1ρa j−1τTM : ∀ j ∈ 2N+ and (ai)

ji=1 ∈ N × N

j−1+

}∪{

τa1ρa2τa3 . . . ρa j−1τa j−1τTM : ∀ j ∈ (2N + 1) and (ai)ji=1 ∈ N × N

j−1+

},

see also Definition 8.5.2, where τTM(0) = 01, τTM(1) = 10 denotes the Thue-Morse substitution. One then says that a substitution Dk,n is associated withσ ∈ Q if σ encodes the rational rotation by k/n. The indicated connection toβ-transformations is established by the following theorem, see Theorem 8.5.8.

Theorem. Let ℓ ≔((ki, ni)

)i∈N with gcd(ki, ni) = 1, let Dki,ni be associated with

σi for all i ∈ N. Set ℓm ≔((ki, ni)

)mi=0 and define for (βm, αm) ∈ Dℓm the map

Tm(x) ≔ {βmx + αm} with Tm-discontinuity γm. Then, for the autocorrelations γTm

of the 1[γm,1)-weighted return time combs with respect to the Parry measure µm andreference point ym, one has

v-limm→∞

γTm = γu,

where u = limm→∞ σ0 . . . σm(1), the subshift Xu is uniquely ergodic and γu denotesthe autocorrelation of u.

Here the autocorrelation γu of u is given by the u-weighted return time combwith respect to the left shift, which is independent of the reference point, as Xu

is uniquely ergodic. In this case u is viewed as a function on the integers, byu : Z→ {0, 1}, z ↦→ uz, z ≥ 0, z ↦→ 0, z < 0. This theorem and the next proposition,see also Proposition 8.5.9, which considers a different case of convergence, firstappeared in [83].

Proposition. For sequences((km, nm)

)m∈N with gcd(km, nm) = 1, let Dkm,nm be as-

sociated with σm for all m ∈ N. Given ym ∈ [0, 1) and (βm, αm) ∈ Dkm,nm , m ∈ N wedefine the map Tm(x) ≔ {βmx+αm} with Tm-discontinuity γm. If limm→∞ km/nm = αand nm → ∞ for m → ∞, the autocorrelations γTm of the 1[γm,1)-weighted returntime combs with respect to the Parry measure µm and reference point ym convergevaguely to

v-limm→∞

γTm = γTα ,


where γTα is the autocorrelation of the rotation Tα by α, given by Ξ(Tα,m|[0,1))(n) =

(1[1−α,1) ∗ ˜1[1−α,1))({αn}) for all n ∈ Z and Ξ(Tα,m|[0,1))(n) = | sin(πn(1−α)

)/(πn)|2

for all n ∈ Z\{0}, Ξ(Tα,m|[0,1))(0) = α2 and ‖Ξ(Tα,m|[0,1))‖ = α.

Figure 1.4: The three sets shown are D1,3, D3,11 and D19,71. The numbers are

related to the continued fraction expansion [0; 3, 1, 2, 1, 4]. The marked posi-

tions in the top-left picture are the points (β1, α1) ≈ (1.1768, 0.2444) ∈ D1,3,

(β2, α2) ≈ (1.0443, 0.2507) ∈ D3,11 and (β3, α3) ≈ (1.0067, 0.2643) ∈ D19,71. The

other graphs are the spectral return measures for (β1, α1), (β2, α2) and (β3, α3) re-

spectively.

By a result of [83] a generalisation has been achieved in Proposition 8.5.11,

which considers fm-weighted return time combs for certain functions fm instead of

1[γm,1)-weighted return time combs. This is specified by the following proposition.

Proposition. For sequences((km, nm)

)m∈N with gcd(km, nm) = 1, let Dkm,nm be as-

sociated with σm for all m ∈ N. Given ym ∈ [0, 1) and (βm, αm) ∈ Dkm,nm, m ∈ Nwe define the map Tm(x) � {βmx + αm} with Parry density μm, where m ∈ N andbounded functions f , fm : [0, 1) → C, where m ∈ N and sup{‖ fm‖∞ : m ∈ N} < ∞.

If limm→∞ km/nm = α and nm → ∞, the set of discontinuities of f has Lebesgue


measure zero and for all δ, ε1, ε2 > 0 there exists an M ∈ N such that

m

⎛⎜⎜⎜⎜⎜⎝⋃m≥M

{z ∈ [0, 1) : sup

x,y∈Bδ(z)| fm(y) − f (x)| ≥ ε2

}⎞⎟⎟⎟⎟⎟⎠ < ε1,

the autocorrelations γTm of the fm-weighted return time combs with respect to theParry measure µm and reference point ym converge to

v-limm→∞ γTm = γTα ,

where γTα is the autocorrelation of a f -weighted return time comb with respect tom and Tα(x) = {x + α}.

This proposition can be combined with return time combs for rigid rotations,if one chooses fm to be Riemann-integrable and non-negative with limm→∞ ∥ fm −

f ∥∞ = 0. In this one can consider (βm, αm) ∈ Dkm,nm ∪{1, km/nm}, where the closureis taken in R2 by adding the single point (1, km/nm), m ∈ N. In this case the map isgiven by Tm(x) = {x + km/nm} and the assumptions of 6.3.10 are fulfilled. A moredetailed explanation is given in Remark 8.5.13.

In a few additional cases more information about the attained vague limit canbe given. One such is if substitutions σi ∈ Q are chosen periodically with periodm ∈ N, one can set σ = σ0 . . . σm being of constant length q ∈ N and Xu = Xσ.If at least one of the contributing substitutions σ0, . . . , σm is not equal to τTM, aresult of Dekking, see [25], can be applied to yield the following proposition, seeProposition 8.5.15.

Proposition. Let ℓ ≔((ki, ni)

)i∈N with gcd(ki, ni) = 1 be a periodic sequence with

period p and at least one tuple being not equal to (1, 2). Let Dki,ni be associatedwith σi for all i ∈ N. Set ℓm ≔

((ki, ni)



of the 1[γm,1)-weighted return time combs with respect to the Parry measure µm andreference point ym, one has that v-limm→∞ γTm = γu, where u = limm→∞ σ

m(1) forσ ≔ σ0 . . . σp of constant length q and Xu = Xσ. Furthermore, ˆγu is a discretemeasure on [0, 1), with its atoms inside the set

⋃n∈N+(q

−nZqn) and ∥ˆγu∥ = k0/n0.

In the case u = limm→∞ τmTM(1), where τTM is associated with D1,2, the subshift

Xu = XτTM is well studied, see e.g. [5, 71, 6, 4]. The spectral return measure ˆγu

has its only atom at 0 and is otherwise given by the vague limit ϱ of the Rieszproducts associated with the Thue-Morse sequence, see Proposition 8.5.16. Thatis somewhat also the case if the first substitution differs, i.e. u = limm→∞ σ0τ

mTM,

where σ0 is associated with Dk,q and is given explicitly in the next theorem, seealso Theorem 8.5.19.


Figure 1.5: The spectral return measure γ given by the Bochner transform for

the autocorrelation of the subshift of a periodic application of σ0σ1, where

σ0 = ρττTM is associated with D3,5 and σ1 = ττTM is associated with D1,3. As

|σ0σ1(0)| = 5 · 3 = 15, the atoms of γ are located in the set {k/15n : 0 ≤ k ≤15n, n ∈ N}. The labelling for the x-axis is done with fifteenth’s and only shows

the numerators.

Theorem. Let gcd(k, q) = 1 and u � limm→∞ στmTM(1) for a σ associated with

Dk,q � D1,2, then

(u � u)∧ = 2−1q−2(1 + g1)(� ◦ s−1

1/q

)∗ δ 1

qZq+ φw,q δ 1

qZq,

where � denotes the Bochner transform of the Thue-Morse substitution, as given inLemma 8.5.18. The map s1/q : [0, 1) → [0, 1/q) is given by x → x/q, the functiong1(x) � cos(2πx) and the density φw,q is defined by

φw,q �1

q2

∣∣∣∣∣∣∣1/2 + e1/2 +∑

0≤r≤q−1,(00w)r=1

er

∣∣∣∣∣∣∣2

δq−1Zq ,

where w � S 2σ(0) = S 2σ(1) and ey(x) = e2πixy. In particular ‖(u � u)∧‖ = k/q.


Figure 1.6: On the left side the spectral return measure for the Thue-Morse sub-

stitution with {0, 1}-alphabet is shown, while on the right side the spectral return

measure for the infinite sequence given by limn→∞ στnTM(1), where σ = ττTM

associated with D1,3 is given. In both cases the atom at zero is taken out of con-

sideration in the picture.

1.2. OUTLINE OF CHAPTERS 17

1.2 Outline of chaptersThis outline provides a short summary of all chapters besides this one. As thiswork brings many different branches of mathematics together, an introductionto every area is given. These chapters and sections will be pointed out in thefollowing, as well as the ones containing the main results of this work.

Ch. 2 Section 2.1 introduces basic concepts for continued fractions which will beused in some of the subsequent proofs. Section 2.2 introduces general nota-tion for symbolic spaces and subshifts. At the end α-repetitive, α-repulsiveand α-finite are defined and it is shown that a subshift is α-repulsive if andonly if it is α-finite in Theorem 2.2.16. In Section 2.3 substitutions τ, ρ areintroduced and are linked to the continued fraction expansion of its rotationnumber via Theorem 2.3.8 such that they encode the orbit of that numberon the unit circle in a unique way. This plays a key role in Chapters 3 and 8.

Ch. 3 This chapter is a collection of results that appeared in [35]. Section 3.1,3.1.1, 3.3.1 and 3.3.2 contain the lemmata and propositions, which are usedto prove the main theorems in the subsequent sections of this chapter. Defi-nition 3.1.3 introduces Sturmian subshifts of slope ξ ∈ (0, 1) which will bestudied in terms of α-repetitive, α-repulsive and α-finite, which are shownto be equivalent in Theorem 3.2.6. In Definition 3.3.2 a spectral metric isdefined for Sturmian subshifts of slope ξ. It turns out in Section 3.3.3 thatthe spectral metric is linked via Theorems 3.3.15 and 3.3.16 to the usualmetric on a subshift by the continued fraction expansion of ξ. When ξ iswell approximable of α-type, see Definition 3.1.1, all notions are equiv-alent, by Theorems 3.2.6 and 3.3.15. Finally the Hausdorff-dimension ofwell-approximable numbers of α-type is calculated in Theorem 3.4.3.

Ch. 4 Here Borel measures and Radon measures are defined as complex-valuedmeasures or non-negative measures. The Riesz theorems used in this workare given in Sections 4.2 and 4.3, while in Section 4.4 measures are decom-posed by the Lebesgue decomposition.

Ch. 5 The notion of Fourier transformation will be given in Definition 5.1.1, aswell as Bochner’s theorem in Section 5.2. Of particular interest for thesubsequent chapters is the space of integers covered in Section 5.2.1.

Ch. 6 Section 6.1 introduces operator theory on a general basis. Sections 6.2and 6.3 present two canonical ways of obtaining positive definite sequences,which have a Bochner transform. These are the so-called spectral measureand spectral return measure respectively and are compared in (6.4). Withreturn combs, a construction will be done that is usually understood as the


autocorrelation of a quasicrystal, see Chapter 7, which is positive definite.With that said Sections 6.4 to 6.6 present an introduction to the spectral the-ory of operators. Special attention should be paid to Section 6.5.1, whichpresents a formula for the spectral return measure of weakly mixing dynam-ical systems in Theorem 6.5.13 in case a Perron-Frobenius operator can bedefined on them.

Ch. 7 The driving factors for this work were aperiodic order and, later, quasicrys-tals. While aperiodic order can be found nearly everywhere throughout thiswork, it turns out that it could be written without any knowledge about qua-sicrystals. The reason is that the definition of a quasicrystal, as a compara-tively young field, is still not fixed and if they are considered in mathematicsthey usually fall into more general structures that had been studied beforequasicrystals were known, such as spectral measures, aperiodic order, Cutand Project Schemes and so on. In this chapter a short overview of qua-sicrystals will be given with special attention to Cut and Project Schemes.

Ch. 8 In Sections 8.1 and 8.2 the parameter space ∆ for β-transformations is intro-duced and Definition 8.2.8 gives the areas in ∆ for which β-transformationsare not weakly mixing by Theorem 8.4.1. The non-mixing regions are fur-ther studied in Section 8.3 in terms of topologically and measure-theoreti-cally conjugacy mappings.

In Section 8.4 these areas are coupled via Definition 8.4.7 by Lemmas 8.3.1and 8.3.3 to attain a representation for the autocorrelation of a return timecomb in Theorem 8.4.10. With an application of Theorem 6.5.13 the spec-tral return measure of said autocorrelation is given by Theorem 8.4.11.It is stated in Corollary 8.4.13 that the spectral return measure of any β-transformation is the sum of spectral return measures of a weakly mixingdynamical system for a β-transformation. In particular the spectral returnmeasure of any β-transformation decomposes into a Lebesgue absolutelycontinuous part and a finite discrete part.

In Section 8.5 sequences of β-transformation given by pairs (β, α) ∈ ∆ forwhich β tends to 1 and α approaches a value in [0, 1) will be studied. Again,special attention is upon the areas in ∆ given by Definition 8.2.8 and Def-inition 8.4.7. All pairs (β, α) will be assumed to belong to one of theseareas and then their combinatorial description is used to establish a link torotations in Lemma 8.5.4. For these β-transformations it is shown in Theo-rem 8.5.8 and Proposition 8.5.9 that the vague limit of their autocorrelationsconverges to the autocorrelation of a subshift given by the combinatorialproperties of said areas. In the subsequent Sections 8.5.1 and 8.5.2 atten-tion is given to certain subshifts that are attained in this way and the results

1.2. OUTLINE OF CHAPTERS 19

are collected in Section 8.6.

The thesis is concluded in three appendices:

A As conjugacies between dynamical systems play an important role in thisthesis, the formalism is presented here with the unit circle as an example.

B Here the canonical choices for the topologies we assume on the functionand functional spaces are introduced. Appendix B.2.2 generalises the cor-respondence between measures and functionals given by Riesz Theorems.

C Appendix C.1 introduces the convolution of functions, measures and func-tionals and states basic properties for them. After that Appendix C.1.1 givesa short introduction to dual spaces and contains a discussion for a straight-forward approach for the existence of the inverse for a Fourier transform.This is generalised in C.2, which introduces the Fourier transform of func-tionals and might be less known than the other fields covered in the ap-pendix. A summary of what will mainly be used in the thesis is given inRemark C.2.4. Finally Appendix C.3 contains some basic properties forFourier transformation, such as the Poisson summation formula.


Chapter 2

Continued fractions and symbolicrepresentation

2.1 Continued fraction expansionThe introduction to continued fractions given in this section uses tool and tech-niques which are mainly described in [51, Ch. 1]. Although the approach isdifferent.

For every ξ ∈ R define r0 ≔ ξ and inductively an ≔ ⌊rn⌋ and rn+1 ≔ (rn−an)−1,as long as rn ∉ Z. This is known as the generalised Euclidean algorithm. Thenrn = an + r−1

n+1 and

ξ = r0 = a0 +1r1= a0 +

1a1 +

1r2

= a0 +1

a1 +1

a2+1r3

= . . . .

One says ξ has continued fraction expansion given by (ai)i∈I(ξ), where I(ξ) ={0, 1, . . . , n} if rn ∈ Z for some n ∈ N and I(ξ) = N otherwise. Notable is theunique choice of all ai, i ∈ I(ξ).

Remark 2.1.1. For the choices above one has r0 ∈ R and a0 ∈ Z, but r0 − a0 =

r0 − ⌊r0⌋ ∈ [0, 1) which implies r1 ∈ (1,∞) and a1 ∈ N+. Inductively that means,for all n ∈ I(ξ) we have rn ∈ (1,∞) and an ∈ N+. In particular the case rn ∈ Nimplies n = max(I(ξ)) and an ∈ N+ ∩ (1,∞), hence an ≥ 2.

The next definition presents a new tool to express ξ in terms of Theorem 2.1.4.In fact they have such an impact on the continued fraction expansion of numbersthat they are interesting in their own right.

Definition 2.1.2. For any sequence (an)n∈I with a0 ∈ Z and an ∈ N+, where eitherI = {0, . . . ,N} is finite or I = N we define p−2 = 0, p−1 = 1 and q−2 = 1, q−1 = 0.

21

22 CHAPTER 2. CONTINUED FRACTIONS AND . . .

Further for any n ∈ I

qn ≔ anqn−1 + qn−2 and pn ≔ an pn−1 + pn−2.

Remark 2.1.3. If a0 = 0 it follows p0 = 0, p1 = 1 and q0 = 1, q1 = a1. For thatreason p0 and q0 are often omitted if a0 = 0 and the sequences (pn)n∈N+ , (qn)n∈N+are considered.

Theorem 2.1.4. For any ξ ∈ R and n ∈ I(ξ)\{sup I(ξ)} we have

ξ =pnrn+1 + pn−1

qnrn+1 + qn−1

Proof. The proof will be done by induction. The start follows directly from thedefinition

ξ = r0 =p−1r0 + p−2

q−1r0 + q−2, r0 = a0 +

1r1=

p0r1 + p−1

q0r1 + q−1.

Whereas the inductive step

pnrn+1 + pn−1

qnrn+1 + qn−1=

pn(an+1 + 1/rn+2) + pn−1

qn(an+1 + 1/rn+2) + qn−1

=an+1 pnrn+2 + pn + pn−1rn+2

an+1qnrn+2 + qn + qn−1rn+2

=(an+1 pn + pn−1)rn+2 + pn

(an+1qn + qn−1)rn+2 + qn=

pn+1rn+2 + pn

qn+1rn+2 + qn,

holds as long as rn+2 is defined, hence n + 2 ∈ I(ξ) is required. �

The equality given in the theorem motivates the following definition

Definition 2.1.5. For any finite sequence (ai)ni=0, where a0 ∈ Z and an ∈ N+ define

[a0; a1, . . . , an] ≔pn

qn.

This definition is a representation, which we will see in Proposition 2.1.9, isan approximation of ξ, that is not explicitly reflected by (pn/qn)n

i=0. The pictureone should keep in mind is the following one for a number ξ, given by

a0 +1

a1 +1

a2+1

a3+1

...+ 1an

≕ ξ =pn−1an + pn−2

qn−1an + qn−2=

pn

qn= [a0; a1, . . . , an], (2.1)

where Theorem 2.1.4 is used to create the link between ξ and pn/qn.

2.1. CONTINUED FRACTION EXPANSION 23

Remark 2.1.6. There is a further subtlety hidden in this definition. While (2.1)works fine with an = 1 and carries over to pn, qn. If ξ ∈ R is such that |I(ξ)| = n <∞ we have an ≥ 2 by Remark 2.1.1. Indeed if an = 1

[a0; a1, . . . , an−1, 1] =a0 +1

a1 +1

a2+1

a3+1

...+ 1an−1+

11

=a0 +1

a1 +1

a2+1

a3+1

...+ 1an−1+1

= [a0; a1, . . . , an−1 + 1].

So while the algorithm yielding (ai)i∈I(ξ) is unique, the representation for continuedfractions via [a0; a1, . . . , an−1, 1] is in general not.

Lemma 2.1.7. For any n ∈ I(ξ) ∪ {−1}

qn pn−1 − pnqn−1 = (−1)n.

Proof. By definition qn = anqn−1+qn−2 and pn = an pn−1+pn−2, which is equivalentto qn pn−1 = an pn−1qn−1 + pn−1qn−2 and pnqn−1 = anqn−1 pn−1 + qn−1 pn−2. Putting thetwo together yields

qn pn−1 − pnqn−1 =an pn−1qn−1 + pn−1qn−2 − (anqn−1 pn−1 + qn−1 pn−2)= − (qn−1 pn−2 − pn−1qn−2).

While q0 p−1 − p0q−1 = 1 · 1 − a0 · 0 = 1 gives the desired formula by induction, italso holds for n = −1, as q−1 p−2 − p−1q−2 = 0 · 0 − 1 · 1 = −1. �

Corollary 2.1.8. For any n ∈ I(ξ)

gcd(pn, qn) = 1.

Proof. For k ≔ gcd(pn, qn) it follows from Lemma 2.1.7

(−1)n

k=

qn

kpn−1 −

pn

kqn−1 ∈ Z.

�

The introduced notions will be categorised in the following Proposition

Proposition 2.1.9. For all n ∈ I(ξ)


(i) Each ξ is approximated by its sequence of (pn/qn)n∈I(ξ)ξ −

pn

qn

≤

1qn+1qn

<1q2

n.

(ii) Two adjacent members of the sequence (pn/qn)n∈I(ξ) are linked by the fol-lowing equality

pn+1

qn+1=

pn

qn+ (−1)n 1

qn+1qn.

(iii) The following chain of inequalities further amplifies the geometric picture

p2n

q2n≤ ξ ≤

p2n+1

q2n+1.

Proof. For the first claim noteξ −

pn

qn

=

pnrn+1 + pn−1

qnrn+1 + qn−1−

pn

qn

=

pnqnrn+1 + qn pn−1

(qnrn+1 + qn−1)qn−

pnqnrn−1 − pnqn−1

(qnrn+1 + qn−1)qn

=

qn pn−1 − pnqn−1

(qn(an+1 + r−1n+2) + qn−1)qn

=

1(an+1qn + qn−1)qn + q2

nr−1n+2

≤ 1

qn+1qn.

The second part is a direct consequence of

qn pn−1 − pnqn−1 = (−1)n ⇔pn−1

qn−1−

pn

qn= (−1)n 1

qnqn−1.

The third claim utilises ξ ∈ [pn/qn − 1/(qn+1qn), pn/qn + 1/(qn+1qn)] from the firstclaim, while one endpoint of the interval is given by the second claim. �

2.2 Symbolic spacesFor a finite alphabet Σ we denote by Σ∗ ≔ {u ∈ Σn : n ∈ N} the set of all finitewords in Σ and by ΣN all infinite words. A semigroup homomorphism σ : Σ→ Σ∗

on Σ∗ or ΣN is called substitution. The name semigroup homomorphism is fromthe fact that σ(u) ↦→ σ(u0)σ(u1)σ(u2) . . . is well-defined, for any finite of infiniteword u. As just indicated finite sequences u = (ui)n−1

i=0 ∈ Σn for some n ∈ N may

2.2. SYMBOLIC SPACES 25

also be denoted by u0u1u2 . . . un−1 or may even be functions u : {0, . . . , n−1} → Σn

and the same goes for infinite sequences. The length of any u ∈ Σ∗ ∪ ΣN is givenby |u| ≔ n, if u = u0 . . . un−1 ∈ Σ

n, while |u| = ∞, if u ∈ ΣN. Moreover for anyv ∈ Σ∗ we define

|u|v ≔ |{n ∈ N : ∀0 ≤ i ≤ |v| − 1, un−i = vi}| ,

to be the occurences of v in u. As already used and known from sequences, lettersof u are adressed by un for 0 ≤ n ≤ |u| − 1, factors of u are finite words of theform u[n,n+m] ≔ (ui)n≤i≤n+m and subwords of u are factors, but may also be infinite,hence of the form u[n,∞]. The prefix of u of length n is defined to be the first nletters of u and is denoted by u|n ≔ u[0,n], while the suffix of length n of u is givenby u[|u|−n,|u|−1] and is only defined for finite words.

Remark 2.2.1. Take note thatN = {0, 1, 2, 3, . . .}, whileN+ = {1, 2, 3, 4, . . .}. Alsowe make use of the convention Σ0 ≔ {∅}, while the empty word is also denoted byε.

Definition 2.2.2. For u ∈ ΣN, v ∈ Σ∗, if it exists, the frequency of v in u is givenby

fv(u) ≔ limn→∞

|u|n|vn.

Further information for the frequency can be found in e.g. [70, Ch. 1.2.4],[71, Ch. 5.3,5.4].

Definition 2.2.3. A word u ∈ ΣN is called periodic, if it exists a v ∈ Σ∗ such thatfor all m ∈ N, u|(m|v|) = vm , where vm ∈ Σm|v| is the unique word that satisfies(vm) j = v( j mod |v|) for 0 ≤ j ≤ m|v|. The word u is called ultimately periodic, if itexists an infinite periodic subword of u. If u is not ultimately periodic it is calledaperiodic.

Definition 2.2.4. Let u ∈ Σ∗ ∪ ΣN. The complexity function p ≔ pu : N → Ncounts all different factors of length n ∈ N in u

p : n ↦→ |{v ∈ Σn : v is a factor of u}| .

Definition 2.2.5. For Σ = {0, 1} a word u ∈ Σ∗ ∪ ΣN is called Sturmian of level min the case m ≔ sup{k ∈ N : p(l + 1) = p(l) + 1 ∀l < k} is a natural number, whilefor m = ∞ it is called Sturmian.

Lemma 2.2.6 ([70, Prp. 1.1.1]). A sequence u is ultimately periodic if and only ifp is bounded, especially there is an n ∈ N such that p(n) ≤ n.


For further details on the complexity function see also [70, Ch. 1.1.2]. Withthat factors of sequences u become interesting in itself and we denote by L(u) ≔{v ∈ Σ∗ : v is a factor of u} the language of u. The language of Sturmian sequencesis special in the sense that for all n ∈ N there is only one word v ∈ Ln(u) ≔ {w ∈L(u) : |u| = n} such that v0, v1 ∈ Ln+1(u); v is then called right special. The setof all right special words is denoted by LR(u) ≔ {v ∈ L(u) : v is right special}.Later we want to see which sequences can be approximated by the language ofan infinite word. The intuitive idea to say that two sequences are equal if they arethe same for arbitrary prefixes can be put into action with the following metric, letu, v ∈ ΣN

d(u, v) ≔ |u ∨ v|−c,

is a metric on ΣN, where u ∨ v ≔ {w ∈ Σ∗ ∪ ΣN : w is a prefix of u and v} denotesthe longest prefix u and v share and 0 < c, ∞−c ≔ 0. It may be extended ontoΣ∗ ∪ ΣN by using that |u ∨ v| = inf{inf {n ∈ N : un ≠ vn}, |u|, |v|}. With that at handa topology can be defined for ΣN via d. Another way to define a topology is givenvia cylinder sets [w] ≔ {u ∈ ΣN : u|n = w} and both ways generate the sametopology on ΣN, which can be seen from [w] = {u ∈ ΣN : d(v, u) � (|w| − 1)−c} forany v ∈ [w] and from {u ∈ ΣN : d(v, u) < (|w| − 1)−c} = {u ∈ ΣN : d(v, u) ≤ |w|−c}

we see that each cylinder is a clopen set and d is an ultra-metric. On this occasionwe would like to introduce one of the most important maps on ΣN, the left shift

S : ΣN → ΣN, (ui)i∈N ↦→ (ui+1)i∈N.

It may also be defined for Σ∗ and in this case we set S (ε) = ε for the empty wordε. As S [w] = [S w] for any w ∈ Σ∗ it is clear that S is continuous.

Definition 2.2.7 (Subshift). A subshift X ⊆ ΣN is a closed shift invariant set, thatis S (X) ⊆ X. The language of a subshift is given by L(X) ≔ {L(u) : u ∈ X} andthe language of right special words is defined by LR(X) ≔ {LR(u) : u ∈ X}. Fora sequence u ∈ ΣN, the subshift of u is the set Xu ≔ {S n(u) : n ∈ N}. Xu may becalled Sturmian subshift (of level n) if u is Sturmian (of level n). Either, Xu or uare called minimal, if every factor of u occurs infinitely often in u with boundedgaps. That is, for every factor v of u exists an rv ∈ N such that for all n ∈ N theword v is a factor of un . . . un+rv . In this case u may also be called recurrent insteadof minimal.

For a minimal subshift Xu it follows for any v ∈ Xu that Xv = Xu and henceL(Xu) = L(u) = L(v) = L(Xv), [70, Prp. 5.1.10]. One can then say that a subshiftX is minimal if Xu = X for all u ∈ X. Furthermore, the complexity function is thesame for all elements of a minimal subshift. In this sense all the information of


a minimal subshift or its language is stored in every of its elements and notionslike complexity and periodicity may be used for the subshift instead of stating thenotion with respect to every element of the subshift.

Remark 2.2.8. In [70, Ch. 6.1] one can find that every Sturmian subshift is min-imal. By definition, the language L(X) of a Sturmian subshift contains a uniqueright special word per length and for any w ∈ LR(X) it follows S k(w) is a rightspecial word for all k ∈ {1, 2, . . . , |w|}.

Definition 2.2.9. For u ∈ ΣN the topological entropy is given by

h ≔ h(u) ≔ limn→∞

log|Σ| (p(n))

n.

Since p is monotone and p(n) ≤ |Σ|n the limit in the topological entropy iswell-defined. Sequences with topological entropy 0 are said to be deterministic.For a primitive substitution ζ (compare Definition 6.6.1), there is a constant C > 0such that p(n) ≤ Cn for all n ∈ N. With that one can deduce that for everyfixed point u of ζ any v ∈ Xu is deterministic, [71, Prp. 5.12, 5.7]. Although thefollowing definition are given for subshifts, they will only find use for minimalsubshifts in this work.

Definition 2.2.10. The repetitive function R : N+ → N+ of a subshift X maps anyn to the smallest n′ such that any element of L(X) with length n′ has all elementsof L(X) with length n as factors.

In other words let X be a subshift, n′ ≔ R(n) and v ∈ L(X), |v| = n′, then forany w ∈ L(X) with |w| = n it follows that w is a factor of v.

Lemma 2.2.11. For any subshift X and u ∈ X, one has pu(n) ≤ R(n) for all n ∈ Nand if X

Proof. Let r = pu(n), then a word w can be constructed, which has r differentfactors of length n. That is |w| = n+ r with factors w[i,i+n], where 0 ≤ i ≤ r − 1 andw[i,i+n] ≠ w[ j, j+n] for i ≠ j. One may believe that w belongs to the shortest wordswith this property and hence R(n) ≥ n + r. �

Definition 2.2.12 (α-repetitive). Let X be a subshift and α ≥ 1, set

Rα ≔ lim supn→∞

R(n)nα

.

X is called α-repetitive if Rα is finite and non-zero.


Take note that if 1 ≤ α < β and 0 < Rβ < ∞, then Rα = ∞. Similarly, if0 < Rα < ∞, then Rβ = 0. The notion α-repetitive has been used before, compare[26] and we remark that the definition given in [26] will never be used in thiswork.

Remark 2.2.13. Let u ∈ ΣN, a subshift X is said to be linearly repetitive or linearlyrecurrent, if and only if, there exists a positive constant C, such that R(n) ≤ Cn, forall n ∈ N+. As aperiodicity of a subshift guarantees that the complexity functionp(n) > n, for all n ∈ N+, Lemma 2.2.6 and p(n) ≤ R(n), Lemma 2.2.11, thisyields that linearly repetitive or linearly recurrent and 1-repetitive are equivalentfor aperiodic subshifts.

Definition 2.2.14 (α-repulsive/-finite). Let X be a subshift and α ≥ 1. Set

ℓα ≔ lim infn→∞

Aα,n,

where for any n ≥ 2

Aα,n ≔ inf{|W | − |w||w|1/α

: w,W ∈ L(X),w is a prefix and

suffix of W, |W | = n and W ≠ w ≠ ∅}.

and if ℓα is finite and non-zero, then we say that X is α-repulsive. For n ≥ 1 set

Q(n) ≔ sup{p ∈ N+ : there exists W ∈ L(X) with |W | = n and W p ∈ L(X)}

and the subshift X is α-finite if the value

Qα ≔ lim supn→∞

Q(n)nα−1

is non-zero and finite. Also, for ease of notation, for a given word v ∈ L(X), welet Q(v) denote the largest integer p such that vp ∈ L(X), in the case that no suchp exists, we set Q(v) ≔ ∞.

Remark 2.2.15. In a similar fashion to α-repetitive, if 1 ≤ α < β and 0 < ℓβ < ∞,then ℓα = 0 and if 0 < Qβ < ∞, then Qα = ∞. Whether for 0 < ℓα < ∞, thenℓβ = ∞ and if 0 < Qα < ∞, then Qβ = 0. To see this it is enough to checkthe properties for ℓα. Suppose that 0 < ℓβ < ∞. Thus, for n ∈ N+ sufficientlylarge, there exist words w,W ∈ L(X) with w a prefix and suffix of W, |W | = n andW ≠ w ≠ ∅, so that

ℓβ

2≤|W | − |w||w|1/β

≤ 2ℓβ.


Hence, |w| ≥ n(2ℓβ + 1)−1, and

ℓβ|w|1/β−1/α

2≤|W | − |w||w|1/α

≤ 2ℓβ|w|1/β−1/α.

Therefore, we have that ℓα = 0.

Theorem 2.2.16 ([35, 28]). For α ≥ 1, we have that X is α-repulsive if and onlyif it is α-finite.

Proof. Let α ≥ 1 be fixed and let X be α-repulsive. Suppose that Qα = ∞. In thiscase there exist sequences of natural numbers (nk)k∈N+ and (pk)k∈N+ satisfying

1. (nk)k∈N+ is increasing with pkn1−αk > k, and

2. there exists W(k) ∈ L(X) with |W(k)| = nk and W pk(k) ∈ L(X).

Thus, we have that pk > 1, for all k sufficiently large. Since W pk−1(k) is a prefix and

a suffix of W pk(k) we have that

|W pk(k)| − |W

pk−1(k) |

|W pk−1(k) |

1/α =|W(k)|

|W(k)|1/α(p(k) − 1)1/α

=nk

nk1/α(pk − 1)1/α ≤

21/αnk(α−1)/α

pk1/α <

21/α

k1/α ,

for all k sufficiently large. Therefore, we have that ℓα = 0.Suppose that Qα = 0. For n ∈ N+ let V(n), v(n) ∈ L(X) be such that |V(n)| = n,

v(n) ≠ V(n) is a prefix and suffix of V(n) and

|V(n)| − |v(n)|

|v(n)|1α

= Aα,n.

Since 0 < ℓα < ∞, this means that there exists a sequence (nk)k∈N+ of naturalnumbers such that 2|v(nk)| > |V(nk)|, for all k ∈ N+. Thus, for each k ∈ N+, thereexists a qk ≥ 2 such that

v(nk) = u(k)u(k) · · · u(k)⏞ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ⏞qk−1

z(k) and V(nk) = u(k)u(k) · · · u(k)⏞ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ⏞qk

z(k),

where u(k), z(k) ∈ L(X) with 0 < |z(k)| < |u(k)|. Hence, it follows that⎛⎜⎜⎜⎜⎜⎝ |V(nk)| − |v(nk)|

|v(nk)|1α

⎞⎟⎟⎟⎟⎟⎠α = (|V(nk)| − |v(nk)|)α

|v(nk)|(2.2)

≥|u(k)|

α

qk|u(k)|=|u(k)|

α−1

qk≥|u(k)|

α−1

Q(u(k))≥|u(k)|

α−1

Q(|u(k)|), (2.3)


where the lengths of the u(k) are unbounded, as otherwise lim supk→∞ Q(u(k)) = ∞.However, since by assumption Qα = 0, we have

lim infn→∞

nα−1

Q(n)= ∞.

This together with (2.2) yields that ℓα = ∞. For the other direction suppose thatQα is non-zero and finite. This means there is a sequence of tuples ((nk, pk))k∈N+so that the sequence (nk)k∈N+ is strictly monotonically increasing such that forthe limit 0 < lim

k→∞pkn1−α

k = Qα < ∞, and for each k ∈ N+ there exists a wordW(k) ∈ L(X) with |W(k)| = nk and

W(k)W(k) · · ·W(k)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞pk

∈ L(X).

For a fixed k ∈ N+, setting

W = W(k)W(k) · · ·W(k)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞pk

and w = W(k)W(k) · · ·W(k)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞pk−1

,

we have that

|W | − |w||w|1/α

=n1−1/α

k

(pk − 1)1/α =

(pk

pk − 1nα−1

k

pk

)1/α

.

This latter value converges to Q−1/αα , and so, we have that ℓα is finite.

By way of contradiction, suppose ℓα = 0. This implies there is a strictlyincreasing sequence of integers (nm)m∈N+ , so that there exist W(nm),w(nm) ∈ L(X)with W(nm) ≠ w(nm), |W(nm)| = nm, w(nm) is a prefix and suffix of W(nm) and

|W(nm)| − |w(nm)|

|w(nm)|1/α <

1m.

This means the two occurrences of w(nm) in W(nm) overlap. Thus, there exist p =pnm ∈ N+ so that

w = u u · · · u⏞ˉ ˉ⏟⏟ˉ ˉ⏞p−1

v and W = u u · · · u⏞ˉ ˉ⏟⏟ˉ ˉ⏞p

v,

where u = u(nm), v = v(nm) ∈ L(X) with 0 < |v| < |u|. Combining the above givesp|u|1−α > mα, and so, Qα = ∞, contradicting the assumption that Qα is finite. �

2.3. SUBSTITUTIONS OF ROTATION TYPE 31

Remark 2.2.17. Let X be a subshift. If α = 1, then a 1-finite subshift is alsocalled power free. X is called repulsive if

ℓ ≔ inf{|W | − |w||w|

:w,W ∈ L(X),w is a prefix and

suffix of W, |W | = n and W ≠ w ≠ ∅}.

strictly larger than zero. It was shown in [46] that power free and repulsive areequivalent and by Theorem 2.2.16 1-repulsive is then equivalent to repulsive.

Proposition 2.2.18. For any α ≥ 1, if a subshift X is α-repulsive, or equivalentlyα-finite, then it is aperiodic.

Proof. We show the contrapositive. Suppose that there exists an ultimately peri-odic v ∈ X with period k ∈ N+. This implies that Q(nk) = ∞, for all n ∈ N+ andso, for all α ≥ 1 we have that Qα = ∞. Therefore, the subshift X is not α-finite forany α ≥ 1. �

Proposition 2.2.19. For an aperiodic subshift X we have that R(n) > nQ(n), forall n ∈ N+.

Proof. Let n ∈ N+ be fixed. Let w ∈ L(X) be such that |w| = n and wQ(n) ∈ L(X).The word wQ(n) has at most n different factors of length n. Thus, since |wQ(n)| =

nQ(n) and since L(X) is aperiodic, we have that R(n) > nQ(n). �

Corollary 2.2.20. For an aperiodic subshift X and for α ≥ 1, we have that Rα ≥

Qα. In particular, Rα = 0 implies Qα = 0 and Qα = ∞ implies Rα = ∞.

2.3 Substitutions of rotation typeThroughout this section consider Σ = {0, 1}, the two letter alphabet. A great dealfor subshifts build from rotations has been done in [70, Ch. 6.3] and [29, Ch.3.2]. Both sources introduce substitutions τ and ρ, see Definition 2.3.1, and relateinfinite sequences generated by them to irrational numbers as it will be done inChapter 3. Although a relation to rational numbers, as given in Theorem 2.3.8 hasbeen suggested in [70] a proof is amiss.

Definition 2.3.1. Throughout this work let τ, ρ, θ and τTM denote the semigrouphomomorphisms on {0, 1}∗, {0, 1}N determined by

τ :

⎧⎪⎪⎨⎪⎪⎩0 ↦→ 01 ↦→ 10

, ρ :

⎧⎪⎪⎨⎪⎪⎩0 ↦→ 011 ↦→ 1

, θ :

⎧⎪⎪⎨⎪⎪⎩0 ↦→ 11 ↦→ 0

, τTM :

⎧⎪⎪⎨⎪⎪⎩0 ↦→ 011 ↦→ 10


Definition 2.3.2. Let (ai)i∈N+ ∈ N × NN+ be a sequence of natural numbers. We

define for l ∈ {0, 1}, L ≔ lθl = lθ(l) the finite words

ωjl ≔ω

jL ≔

⎧⎪⎪⎨⎪⎪⎩τa1ρa2τa3 . . . τa j−1ρa j−1(L), (−1) j = 1τa1ρa2τa3 . . . ρa j−1τa j−1(L), (−1) j = −1

ω00 ≔0, ω0

1 ≔ 1, ω1l ≔ ω1

L ≔ L,

where j ≥ 2.

Remark 2.3.3. Some consequences of Definition 2.3.2 are

L = lθl = τTMl =

⎧⎪⎪⎨⎪⎪⎩01, l = 010, l = 1

.

and the cases (−1) j ∈ {−1, 1} just gives off if j is even or odd. As (ai)i∈N+ ∈ N×NN+ ,

the choice a0 = 0 is within the definition, while all other ai ≥ 1 for i ≥ 2.

Example 2.3.4. Let n ∈ N,

ρn(01) =01n+1, τn(01) =010n = ω10,

ρn(10) =101n, τn(10) =10n+1 = ω11.

A direct consequence of the former observations are the identities ρτθ = ρθρ =θτρ. These obervations are expressed by the following diagram for all n ∈ N

τn(10) ρn(01)

τn(01) ρn(10)

θ

01S 2 10S 2

θ

10S 2 01S 2

We conclude this example by noting down ω2l as a mixture of ω’s

ω20 =τ

a1(01a2) = 0(10a1)a2 = ω00(ω1

1)a2 ,

ω21 =τ

a1(101a2−1) = 10a10(10a1)a2−1 = ω11ω

00(ω1

1)a2−1.

This observation holds in general and will be discussed in the upcoming lemma.

Lemma 2.3.5. Let j ≥ 2 and l ∈ {0, 1}, then ω jl can also be expressed by one of

the following cases:

j even:

ωjl =

⎧⎪⎪⎨⎪⎪⎩ω j−20

(ω

j−11

)a j , l = 0ω

j−11 ω

j−20

(ω

j−11

)a j−1, l = 1


j odd:

ωjl =

⎧⎪⎪⎨⎪⎪⎩ω j−10 ω

j−21

(ω

j−10

)a j−1, l = 0

ωj−21

(ω

j−10

)a j , l = 1

Proof. The proof is done by induction. In fact Example 2.3.4 shows the base caseof the induction, whether in the following the inductive step is given by using thecalculations done in the previous example.

For an even j that is:

ωjl =

⎧⎪⎪⎨⎪⎪⎩τa1ρa2 . . . τa j−1(01a j), l = 0τa1ρa2 . . . τa j−1(101a j−1), l = 1

=

⎧⎪⎪⎨⎪⎪⎩τa1ρa2 . . . ρa j−2−1(01)(τa1ρa2 . . . τa j−1−1(10)

)a j , l = 0τa1ρa2 . . . τa j−1−1(10)τa1ρa2 . . . ρa j−2−1(01)

(τa1ρa2 . . . τa j−1−1(10)

)a j−1, l = 1

=

⎧⎪⎪⎨⎪⎪⎩ω j−20

(ω

j−11

)a j , l = 0ω

j−11 ω

j−20

(ω

j−11

)a j−1, l = 1

While an odd j gives:

ωjl =

⎧⎪⎪⎨⎪⎪⎩τa1ρa2 . . . ρa j−1(010a j−1), l = 0τa1ρa2 . . . ρa j−1(10a j), l = 1

=

⎧⎪⎪⎨⎪⎪⎩τa1ρa2 . . . ρa j−1−1(01)τa1ρa2 . . . τa j−2−1(10)(τa1ρa2 . . . ρa j−1−1(01)

)a j−1, l = 0

τa1ρa2 . . . τa j−2−1(10)(τa1ρa2 . . . ρa j−1−1(01)

)a j , l = 1

=

⎧⎪⎪⎨⎪⎪⎩ω j−10 ω

j−21

(ω

j−10

)a j−1, l = 0

ωj−21

(ω

j−10

)a j , l = 1

�

We conclude this section by showing the relation diagram between ωjl for

different choices of l.

Corollary 2.3.6. For all j ∈ N+ the following diagram commutes

ωj1 θω

j1

ωj0 θω

j0

θ

01S 2 10S 2

θ

10S 2 01S 2


Proof. The proof is a straightforward inductive application of the diagram shownin Example 2.3.4. �

Definition 2.3.7 (Rational rotation). Let p, q ∈ N+, gcd(p, q) = 1, p/q ∈ (0, 1).The finite sequence κp/q

0 , κp/q1 ∈ {0, 1}q given by

κp/q0 ( j)≔1{1,...,p}( jp mod q)=1(0, p

q ]

(j p

q mod 1)=1(1− p

q ,1)∪{0}

((1 − p

q

)+ j p

q mod 1),

κp/q1 ( j)≔1{0,...,p−1}( jp mod q)=1[0, p

q )

(j p

q mod 1)=1[1− p

q ,1)

((1 − p

q

)+ j p

q mod 1),

where j ∈ {0, . . . , q − 1} encodes the orbit of p modulo q or p/q modulo 1.

The next theorem connects rotation to substitutions. The assumption for anirrational is done to provide a clearer statement.

Theorem 2.3.8. Let [0; a1 + 1, a2, . . .] ∈ (0, 1) denote an irrational number. Forany n ∈ N the following holds

if n is even:

κpn/qn1 =τa1ρa2 · · · ρan−1(10), (2.4)

if n is odd:

κpn/qn1 =τa1ρa2 · · · τan−1(10), (2.5)

Proof. The prove will be an induction with respect to the continued fraction en-tries. Suppose n = 1, then p1 = 1, q1 = a1 + 1, which implies

κ1/(a1+1)1 = 1 0 · · · 0⏞⏟⏟⏞

a1−times

= τa1−1(10).

Set

κmj ≔ κ

pm/qm1 ( j) = 1[0, pm

qm)

(j pm

qmmod 1

),

where j ∈ {0, . . . , qm − 1},m ≥ 1 and assume that the statement holds for n − 1.An application of Proposition 2.1.9 (ii) yields(

qn−1pnqn

mod 1)=

(qn−1

(pn−1qn−1+ (−1)n−1 1

qnqn−1

)mod 1

)=

((−1)n−1 1

qnmod 1

)∈

[0, pn

qn

)if and only if n is odd, while

1[0, pn−1qn−1

)

(qn−1

pn−1qn−1

mod 1)=1,


regardless of n being even or odd. Another application of Proposition 2.1.9 (ii) letus determine the distance between κn−1

0 = 1 − pn−1/qn−1 and κn0 = 1 − pn/qn to be

exactly 1/(qnqn−1). Thus, for n being odd, the orbits induced by κn−1 and κn matchas long as (

j pnqn

mod 1)−

(j pn−1

qn−1mod 1

)=

jqnqn−1

≤1

qn−1, (2.6)

where the absolute value is taken with respect to the unit circle and holds as longas j ∈ {0, . . . , qn − 1}. If n is even, then one can see that the backward orbit has thesame properties we have just shown for the (forward) orbit in the case of n beingodd. Further, if n is odd

κn = κn−1 · · · κn−1⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞an−times

r (2.7)

where r ∈ {0, 1}qn−2 . By taking the backward orbit and (2.6) into account κn−1 is asuffix of κn, hence r is a suffix of κn−1. Therefore

κn−1 = τa1ρa2 · · · ρan−1−1(10)= τa1ρa2 · · · τan−2(10 1 · · · 1⏞⏟⏟⏞

(an−1−1)−times

)

= τa1ρa2 · · · τan−2−1(100 10 · · · 10⏞ˉ ˉ⏟⏟ˉ ˉ⏞(an−1−1)−times

).

shows r = τa1ρa2 · · · τan−2−1(10) (note that in the n = 2 one has r = 0) and it followstogether with (2.5) and (2.7) that

κn = τa1ρa2 · · · ρan−1−1(10) · · · τa1ρa2 · · · ρan−1−1(10)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞an−times

τa1ρa2 · · · τan−2−1(10)

As ρ(1) = 1, one can conclude

κn =τa1ρa2 · · · ρan−1−1(1) τa1ρa2 · · · ρan−1−1(01) · · · τa1ρa2 · · · ρan−1−1(01)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞(an−1)−times

τa1ρa2 · · · ρan−1−1(0)τa1ρa2 · · · τan−2−1(10)=τa1ρa2 · · · ρan−1(1) τa1ρa2 · · · ρan−1(0) · · · τa1ρa2 · · · ρan−1(0)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞

(an−1)−times

τa1ρa2 · · · ρan−1−1(0)τa1ρa2 · · · τan−2ρan−1−1(1)=τa1ρa2 · · · ρan−1(1) τa1ρa2 · · · ρan−1(0) · · · τa1ρa2 · · · ρan−1(0)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞

(an−1)−times

τa1ρa2 · · · ρan−1(0)


=τa1ρa2 · · · ρan−1(1) τa1ρa2 . . . ρan−1(0) · · · τa1ρa2 · · · ρan−1(0)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞an−times

=τa1ρa2 · · · ρan−1τan−1(10).

If n is even, first the backward orbit is taken to deduce

κn = r κn−1 · · · κn−1⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞an−times

where r ∈ {0, 1}qn−2 . Now, one takes the (forward) orbit to obtain that κn−1 is aprefix of κn and hence

κn = κn−1r′ κn−1 · · · κn−1⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞(an−1)−times

, (2.8)

where r′ ∈ {0, 1}qn−2 is a suffix of κn−1. With that in mind the equality

κn−1 = τa1ρa2 · · · τan−1−1(10) = τa1ρa2 · · · ρan−2(1 0 · · · 0⏞⏟⏟⏞(an−1)−times

)

= τa1ρa2 · · · ρan−2−1(1 01 · · · 01⏞ˉ ˉ⏟⏟ˉ ˉ⏞(an−1)−times

)

implies r′ = τa1ρa2 · · · ρan−2−1(01). This equality together with (2.4) and (2.8) yields

κn =τa1ρa2 · · · τan−1−1(10)τa1ρa2 · · · ρan−2−1(01)

τa1ρa2 · · · τan−1−1(10) · · · τa1ρa2 · · · τan−1−1(10)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞(an−1)−times

.

Finally by τ(0) = 0 and ρ(1) = 1 one obtains

κn = τa1ρa2 · · · τan−1(1)τa1ρa2 · · · ρan−2(0) τa1ρa2 · · · τan−1(1) · · · τa1ρa2 · · · τan−1(1)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞(an−1)−times

= τa1ρa2 · · · τan−1(1)τa1ρa2 · · · τan−1(0) τa1ρa2 · · · τan−1(1) · · · τa1ρa2 · · · τan−1(1)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞(an−1)−times

= τa1ρa2 · · · τan−1(10 1 · · · 1⏞⏟⏟⏞(an−1)−times

)

= τa1ρa2 · · · ρan−1(10).

�

As an immediate consequence of the theorem we note down.


Corollary 2.3.9. Let n ∈ N+, pn/qn = [0; a1 + 1, . . . , an] ∈ (0, 1). Then

κpn/qnl = ωn

l =

⎧⎪⎪⎨⎪⎪⎩τa1ρa2τa3 . . . τan−1ρan−1(lθl), n evenτa1ρa2τa3 . . . ρan−1τan−1(lθl), n odd

whereas l ∈ {0, 1}.

Proof. The only thing that remains to check is the case l = 0, which is due toωn

0 = 01S 2ωn1. �

Example 2.3.10. For x = [0; a1] one has

(ωn1)k = 1[0,0] (k mod an) = 10a1−1 = τa1−1(10).

For 7/16 = [0; 2, 3, 2] one has

τ1ρ3τ1(10) = 1001010100101010 =(1[0,7)(7k mod 16)

)k∈Z16

.

Remark 2.3.11. For a continued fraction pn/qn = [0; a1 + 1, . . . , an] ∈ (0, 1),the cases a0 = 0 or an = 1 are eligible choices with regards to Corollary 2.3.9.Further the critical choice [0; 1] = 1 is not included and [0; 2] = 1/2 correspondsto τ1−1(lθl) = τTM(l). An alternate version of Corollary 2.3.9 is for n ∈ N+ to letpn/qn = [0; a1, . . . , an] ∈ (0, 1), where (ai)n

i=1 ∈ Nn+. Then

κpn/qnl = ωn

l =

⎧⎪⎪⎨⎪⎪⎩τa1−1ρa2τa3 . . . τan−1ρan−1(lθl), n evenτa1−1ρa2τa3 . . . ρan−1τan−1(lθl), n odd

whereas l ∈ {0, 1}.


Chapter 3

Holder regularity for irrationalnumbers and their subshifts

In this chapter the article [35] will be revisited. It investigates the three com-binatorial properties α-repetitive, α-repulsive and α-finite for Sturmian subshiftscorresponding to an irrational number. They are used to describe repeating pat-terns for these subshifts and are connected to the continued fraction expansionof the corresponding irrational number. A setting outside of the one of Sturmiansequences for the three combinatorical properties (α-repetitive, α-repulsive and α-finite) is not considered in this work but can be seen in [28], jointly with Dreher,Keßebohmer, Samuel and Steffens, for Griorchuk sequences and examples aregiven which demonstrate that the notions of α-repetitive and α-repulsive are infact different.

3.1 Bounds on continued fraction expansionThroughout this chapter ξ = [0; 1, a2, a3, . . . ] ∈ [0, 1] will denote an irrationalnumber and pn = pn(ξ), qn = qn(ξ) are associated with its continuous fractionexpansion, see Section 2.1 for an introduction.

Definition 3.1.1. For α ≥ 1 and an irrational number ξ ∈ [0, 1], set

Aα(ξ) ≔ lim supn→∞

anq1−αn−1

and define

Θα≔ {ξ ∈ [0, 1] : 0 < Aα(ξ)}, Θα ≔ {ξ ∈ [0, 1] : Aα(ξ) < ∞}, Θα ≔ Θα ∩ Θα.

Further, we say that ξ is

39

40 CHAPTER 3. HOLDER REGULARITY FOR IRRATIONAL. . .

1. well-approximable of α-type, if Aα(ξ) < ∞,

2. well-approximable of α-type, if Aα(ξ) > 0,

3. well-approximable of α-type, if 0 < Aα(ξ) < ∞.

Notice, any irrational ξ ∈ [0, 1] is well-approximable of 1-type. Further, thecondition that an irrational ξ ∈ [0, 1] is well-approximable of 1-type, and hence of1-type, is equivalent to the continued fraction entries of ξ being bounded. Further,by construction Θα ∩ Θβ = ∅ for all α, β ≥ 1, α ≠ β, as if 0 ≤ Aα(ξ) < ∞, thenfor a subsequence (nk)k∈N it exists an N ∈ N such that anq1−(α+ε)

n−1 ≤ q−εn−y(Aα(ξ) + ε)and respectively ankq

1−(α−ε)nk−1 ≥ qεnk−y(Aα(ξ) − ε) for all k, n ≥ N. We observe that if

a1 = 1, r ≔ [a2; a3, a4, . . .], then

1 − [0; 1, a2, . . .] = 1 −1

1 + 1r

=1 + r1 + r

−r

r + 1=

11 + r

= [0; 1 + a2, a3, . . .]. (3.1)

Therefore any ξ = [0; 1, a2, a3, . . . ] ≥ 1/2 has a corresponding number 1 − ξ =[0; a2 + 1, a3, . . . ] ≤ 1/2 in the sense that qn(ξ) = qn−1(1 − ξ) for all n ∈ N+.

Proposition 3.1.2. For an irrational ξ, we have that

1. ξ is well-approximable of α-type if and only if 1− ξ is well-approximable ofα-type, and

2. ξ is well-approximable of α-type if and only if 1− ξ is well-approximable ofα-type.

Proof. This is a consequence of (3.1) and Definition 3.1.1. �

3.1.1 Sturmian subshifts of slope ξDefinition 3.1.3. Let ξ ∈ (0, 1) be an irrational number with continued fractionexpansion [0; a1 + 1, a2, a3, . . .] = ξ. Set x ≔ limn→∞ τ

a1ρa2τa3 . . . ρa2n−1τTM(0) =limn→∞ ω

n0. The subshift Xx is then called Sturmian subshift of slope ξ.

The mentioned limit in the definition always exists, as ωn0 is a prefix of ωn+1

0for all n ∈ N, where ωn

0 is chosen according to Definition 2.3.1. Note that the therepetitive function R is called recurrency function in [63].

Theorem 3.1.4 ([63, Thm. 10.1]). For Xx given via the continued fraction expan-sion of ξ ∈ (0, 1), we have that

R(n) =

⎧⎪⎪⎨⎪⎪⎩R(n − 1) + 1 if n ∈ N+ \ {qk}k∈N+ ,

qk+1 + 2qk − 1 if n = qk for some k ∈ N+.

3.2. RIGHT SPECIAL FACTORS IN STURMIAN SUBSHIFTS 41

Theorem 3.1.5. A Sturmian subshift of slope ξ is a Sturmian subshift. Further-more it is aperiodic and minimal with respect to the left shift S .

Proof. Minimality is due to Theorem 3.1.4, as a repetitive function that does notattain the value infinity implies that every factor of x = limn→∞ ω

n0 occurs in-

finitely often with bounded gaps, which then implies that Xx is minimal, by Def-inition 2.2.7. That the subshift is Sturmian can be found in [70, Ch. 6.1.2] orfollows from minimality togehter with the fact that the language is Sturmian oflevel n for all n ∈ N+, which can be found after Definition 8.2.1 and is then linkedto our language via Lemma 8.5.1 and Theorem 2.3.8. Aperiodicity is then clearby Lemma 2.2.6, since p(n + 1) = p(n) + 1. �

A natural question at this point is if every Sturmian subshift X can be given bya Sturmian subshift of slope ξ for some ξ ∈ (0, 1) and in [70, Ch. 6] it is shownthat this is indeed the case (A comment on that can be found in [70] betweenProposition 6.1.17 and Exercise 6.1.18).

Remark 3.1.6. For an irrational ξ = [0; 1, a2, a3, . . .] one has for any n ∈ N+,(ai)n

i=1 with a1 = 1 by Example 2.3.4

θτ0ρa2τa3 . . . τTM(0) = τa2ρa3 . . . θτTM(0) = τa2ρa3 . . . τTM(1)

= ωn1 = 10S 2ωn

0, (3.2)

where the last equality is by Corollary 2.3.6. As both subshifts induced by ω1 ≔limn→∞ ω

n1 and ω0 ≔ limn→∞ ω

n0 are minimal it follows L(ω1) = L(10S 2ω0) =

L(ω0) and therefore Xω0 = Xω1 . From (3.2) one also has that ω1 and hence ω0 areassociated with the continued fraction expansion [0; a2 + 1, a3, a4], which equals1 − ξ by (3.1). Setting x ≔ limn→∞ τ

0ρa2τa3 . . . ρ2n−1τTM(0) the subshift Xx associ-ated with ξ is equal to Xθ(ω0), the subshift associated with 1 − ξ.

In scope of the previous remark it will always be assumed that an irrationalnumber ξ ∈ (0, 1/2) from here on onwards. Furthermore, as the subshift Xx asso-ciated with x given via the continued fraction expansion of ξ ∈ (0, 1) is minimal,it is often referred to by X.

3.2 Right special factors in Sturmian subshifts

Before giving the definition of a spectral metric a further investigation of LR(X),the set of all right special words of a Sturmian subshift X is desired and will bedone in this section.


Definition 3.2.1. For ξ = [0; a1 + 1, a2, . . . ] ∈ (0, 1) irrational, where (ai)i∈N ∈

N × NN+ set

R0 = R0(ξ) ≔ (0), L0 = L0(ξ) ≔ (1)Rk = Rk(ξ) ≔ τa1ρa2τa3ρa4 · · · τa2k−1ρa2k(0),Lk = Lk(ξ) ≔ τa1ρa2τa3ρa4 · · · τa2k−1ρa2k(1)

for any k ∈ N.

The former definition is consistent with our current understanding of a se-quence induced by an irrational number, as from Definition 2.3.2 we have (−1)2k =

1 and hence

ω2k0 =τ

a1ρa2τa3 . . . τa j−1ρa2k−1(01)=τa1ρa2τa3 . . . τa j−1ρa2k(0)=Rk

and

ω2k1 =τ

a1ρa2τa3 . . . τa j−1ρa2k−1(10)

=τa1ρa2τa3 . . . τa j−1ρa2k−1(1) τa1ρa2τa3 . . . τa j−1ρa2k−1(0)

=τa1ρa2τa3 . . . τa j−1ρa2k(1) τa1ρa2τa3 . . . τa j−1ρa2k−1(0)

=Lk τa1ρa2τa3 . . . τa j−1ρa2k−1(0).

As limn→∞ ωnl exists for l ∈ {0, 1} one can yield convergence of limk→∞ Rk =

limk→∞ ω2k0 and limk→∞Lk = limk→∞ ω

2k1 . There is also a recursive representation

of Rk and Lk which will mainly be used form here on onwards.

Proposition 3.2.2. Let ξ = [0; a1+1, a2, . . . ] ∈ (0, 1) be an irrational number. Forall k ∈ N+ we have

|Rk| = q2k,

Rk = Rk−1Lk . . .Lk⏞ˉ ˉ⏟⏟ˉ ˉ⏞a2k

,

|Lk| = q2k−1,

Lk = Lk−1 Rk−1 . . .Rk−1⏞ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ⏞a2k−1

.

Proof. For Rk this is exactly Lemma 2.3.5 for j = 2k and l = 0 by the previousdiscussion, whether for Lk one has j = 2k, l = 1. �

Corollary 3.2.3. Let ξ = [0; a1 + 1, a2, . . . ] ∈ (0, 1) be irrational, k ∈ N+, n ∈{0, 1, . . . , a2(k+1) − 1} and m ∈ {0, 1, . . . , a2(k+1)−1 − 1}. The words

RkLk+1 . . .Lk+1⏞ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ⏞n

and Lk+1 Rk . . .Rk⏞ˉ ˉ⏟⏟ˉ ˉ⏞m

are right special.


Proof. That they are right special is due to Proposition 3.2.2, as (Rk)0 = 0 and(Lk)0 = 1 for all k ∈ N. For the remaining part observe

S |Lk |+a2(k+1)−1 |Rk |+(a2(k+1)−(n+1))|Lk+1 |(Rk+1) = RkLk+1 . . .Lk+1⏞ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ⏞n

,

S |Rk−1 |+a2k |Lk |+(a2(k+1)−1−(m+1))|Rk |(Lk+1) = Lk Rk . . .Rk⏞ˉ ˉ⏟⏟ˉ ˉ⏞m

.

�

Remark 3.2.4. For a Sturmian subshift X of slope ξ the sequences x, y will usuallydenote the unique infinite words with x||Rk |= Rk and y||Lk |= Lk, for all k ∈ N+. Asx = limn→∞ τ

a1ρa2τa3 . . . ρ2n−1(01) and (S |Lk−1 |Lk)||Rk−1 | = Rk−1 for all k ∈ N+, bothx, y ∈ X and hence, by minimality, X = Xx = Xy.

With that the notions α-repetetive, α-repulsive and α-finite for α > 1 can beconnected to the factors Rk and Lk, k ∈ N, of a Sturmian subshift of slope ξand the following theorem, preceded by a remark, establishes link a if ξ well-approximable of α-type.

Remark 3.2.5. An analogue of Theorem 3.2.6 for α = 1 of a Sturmian subshiftof slope ξ is given in [47] in Lemma 4.9 via Remark 2.2.17. It states that 1-repulsive (or equivalent 1-finite) is equivalent to the continued fraction expansionof ξ being bounded. Thus Theorem 3.2.6 can be seen to deal with the case, whenthe continued fraction expansion of ξ is unbounded.

Theorem 3.2.6. For α > 1 and ξ ∈ [0, 1] irrational, the following are equivalent.

1. The Sturmian subshift of slope ξ is α-repetitive.

2. The Sturmian subshift of slope ξ is α-repulsive.

3. The Sturmian subshift of slope ξ is α-finite.

4. The Sturmian subshift of slope ξ is well-approximable of α-type, i.e. ξ ∈ Θα.

The proof of Theorem 3.2.6 is divided into the following implications: 1⇒ 2⇒ 4, 4⇒ 1, 4⇒ 3. Note that 3⇔ 2 is due to Theorem 2.2.16.

Proof of Theorem 3.2.6. 1⇒ 2: Assume that the statement is false, in which caseeither ℓα = 0 or ℓα = ∞. First we consider the case ℓα = 0. By definition of ℓα,there exist words W,w ∈ L(X) such that w is a prefix and suffix of W, W ≠ w ≠ ∅and

1 ≤ |W | − |w| ≤⌊|w|1/α

21/αR1/αα

⌋and R(n) ≤ 2Rαnα, (3.3)


for all n ≥ |w|. Further, for all i ∈ {1, 2, . . . , |w|}, we have that

wi = Wi = Wi+|W |−|w|, (3.4)

where we recall that wk and Wk respectively denote the k-th letter of w and W. Bythe property of α-repetitive, for all words u ∈ L(X) with

|u| =⌊|w|1/α

21/αR1/αα

⌋,

we have that u is a factor of w. In particular, letting ξ ∈ X and k ∈ N+, the factor(ξk, ξk+1, . . . , ξk+⌊|w|1/α2−1/αR−1/α

α ⌋

),

of ξ is a factor of w. This together with (3.3) and (3.4) yields that ξk = ξk+|W |−|w| forall k ∈ N+, and thus, ξ is periodic. This contradicts the aperiodicity and minimalityof X. Therefore, if X is α-repetitive and not α-repulsive, then ℓα = ∞. For ease ofnotation set Bk = inf{Aα,n : n ≥ akqk−1}. By Proposition 3.2.2, for all k ∈ N+ wehave that

W ≔ Lk . . .Lk⏞ˉ ˉ⏟⏟ˉ ˉ⏞a2k

, w ≔ Lk . . .Lk⏞ˉ ˉ⏟⏟ˉ ˉ⏞a2k−1

, W ′ ≔ Rk−1 . . .Rk−1⏞ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ⏞a2k−1

, w′ ≔ Rk−1 . . .Rk−1⏞ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ⏞a2k−1−1

(3.5)

all belong to the language L(X) and

|W | − |w||w|1/α

=|Lk|

1−1/α

(a2k − 1)1/α =q1−1/α

2k−1

(a2k − 1)1/α ,

provided that a2k ≠ 1. In the same manner

|W ′| − |w′||w′|1/α

=|Rk−1|

1−1/α

a2k−1 − 1=

q1−1/α2(k−1)

(a2k−1 − 1)1/α ,

provided that a2k−1 ≠ 1. Hence, for k ∈ N+ with ak ≠ 1,

Bk ≤ q1−1/αk−1 (ak − 1)−1/α. (3.6)

Thus, since by assumption ℓα = ∞, since Bk ≤ Bk+1, for all k ∈ N+, and since(qk)k∈N+ is an unbounded monotonic sequence, given N ∈ N+ there exists M ∈ N+so that a jq1−α

j−1 < N−α, for all j ≥ M. For all n ∈ N+ let m(n) be the largest naturalnumber so that qm(n) ≤ n. By Theorem 3.1.4, for all n ∈ N+, so that m(n) ≥ M,

R(n)nα≤

qm(n)+1 + 2qm(n) − 1 + qm(n)+1 − qm(n)

nα


≤2am(n)+1qm(n) + 2qm(n)−1 + qm(n)

qαm(n)

≤2

Nα+

2qm(n)−1

qαm(n)

+qm(n)

qαm(n)

.

Hence, we have that Rα ≤ 2N−α. However, N was chosen arbitrary and so Rα = 0,this contradicts the initial assumption that X is α-repetitive.

2 ⇒ 4: Let [0; a1 + 1, a2, . . . ] denote the continued fraction expansion ofξ. Since the Sturmian subshift X is α-repulsive and α > 1 we have that thecontinued fraction entries of ξ are unbounded, as it is mentioned in the latterpart of Remark 3.2.5 that the continued fraction entries of ξ are bounded if andonly if α = 1. In particular, infinitely often we have that an ≠ 1. SettingBk = inf{Aα,n : n ≥ akqk−1}, as in (3.6), we have that Bk ≤ q1−1/α

k−1 (ak − 1)−1/α,for all k ∈ N+ with ak ≠ 1. Since Bk ≤ Bk+1, there exists N ∈ N+ so that,2α/ℓαα ≥ (an − 1)q1−α

n−1 , for all n ≥ N with an ≠ 1. Hence, since the sequence(qn)n∈N+ is an unbounded monotonic sequence and since, X is α-repulsive,

Aα(ξ) = lim supn→∞

anq1−αn−1 ≤

2α

ℓαα< ∞.

It remains is to show that Aα(ξ) > 0. We have observed that if the Sturmiansubshift X is α-repulsive, then the continued fraction entries of ξ are unbounded.In particular, infinitely often we have that an ≠ 1. Thus, letting W,w,W ′,w′ beas in (3.5), if Aα(ξ) = 0, then Bk = 0, for all k ∈ N+, and hence ℓα = 0. Thiscontradicts the assumption that X is α-repulsive. Hence, if the Sturmian subshiftX is α-repulsive, then Aα(ξ) > 0.

4 ⇒ 1: Let m(n) denotes the largest integer so that qm(n) < n. Since Aα(ξ) <∞, there exists a constant c > 1 so that am+1 ≤ cqα−1

m , for all m ∈ N+. ByTheorem 3.1.4 and the recursive definition of the sequence (qn)n∈N+ , we have forall n ∈ N+,

R(n) ≤ R(qm(n)) + am(n)+1qm(n)

= 2am(n)+1qm(n) + qm(n)−1 + 2qm(n) − 1≤ 2cqαm(n)

+ qm(n)−1 + 2qm(n)

≤ (2c + 3)nα.

In particular, if ξ is well-approximable of α-type then Rα is finite. Further, by The-orem 3.1.4, the recursive definition of the sequence (qn)n∈N+ and the assumptionthat Aα(ξ) > 0, we have that

Rα ≥ lim supk∈N+

R(qk)qαk= lim sup

k∈N+

qk+1 + 2qk − 1qαk

≥ lim supk∈N+

ak+1qk

qαk= Aα(ξ) > 0.

That is, if ξ is well-approximable of α-type, then 0 < Rα.


4⇒ 3: By Proposition 3.2.2 and the definition of Q(n), we have Q(qn) ≥ an+1

and so

Qα = lim supn→∞

Q(n)nα−1 ≥ lim sup

n→∞

Q(qn)qα−1

n≥ lim sup

n→∞

an+1

qα−1n= Aα(ξ) > 0.

Thus, if ξ is well-approximable of α-type and X was not α-finite, then Qα wouldbe infinite. By way of contradiction assume that ξ is well-approximable of α-typeand X and that Qα = ∞. This means there exists a sequence of tuples ((nk, pk))k∈N+of natural numbers such that the sequences (nk)k∈N+ and (pk)k∈N+ are strictly in-creasing and lim

n→∞pkn1−α

k = ∞ and for each k ∈ N+ there exists a word W(k) ∈ L(X)with |W(k)| = nk and W(k)W(k) · · ·W(k)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞

pk

∈ L(X). For a fixed k ∈ N+, setting

W = W(k)W(k) · · ·W(k)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞pk

and w = W(k)W(k) · · ·W(k)⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞pk−1

,

we have

|W | − |w||w|1/α

=n1−1/α

k

(pk − 1)1/α =

(pk

pk − 1nα−1

k

pk

)1/α

=

(pk

pk − 1

(pkn1−α

k

)−1)1/α

.

This latter value converges to zero as k increases to infinity. Therefore, ℓα = 0 andso X is not α-repulsive. This is a contradiction as have we already seen that ξ iswell-approximable of α-type if and only if X is α-repulsive. �

As Sturmian words are characterised by their right special factors we will paymore attention to them in the following.

Definition 3.2.7. Let u ∈ X, we define the function

bn(u) =

⎧⎪⎪⎨⎪⎪⎩1 if u|n is a right special word,0 otherwise,

for all n ∈ N+.

Corollary 3.2.8. Let ξ = [0; a1+1, a2, . . . ] ∈ [0, 1/2] and let X denote a Sturmiansubshift of slope ξ. If x, y ∈ X are the unique infinite words such that x||Rm |= Rm

and y||Lm |= Lm, for all m ∈ N+, then

1. bn(x) = 1 if and only if n = jq2k−1 + q2k−2 for some k ∈ N+ and somej ∈ {0, 1, . . . , a2k − 1}, and

2. bm(y) = 1 if and only if m = iq2l + q2l−1 for some l ∈ N+ and some i ∈{0, 1, . . . , a2l+1 − 1}.


Proof. Corollary 3.2.3 gives the reverse implication: If n = jq2k−1+q2k−2, for somek ∈ N+ and some j ∈ {0, 1, . . . , a2k − 1}, then bn(x) = 1, and if m = iq2l + q2l−1, forsome l ∈ N+ and i ∈ {0, 1, . . . , a2l+1 − 1}, then bm(y) = 1.

For the forward implication, we show the result for bn(x) and bm(y) wheren ≤ |R1| = q2 and where m ≤ |L2| = q3 after which we proceed by induction toobtain the general result.

By Remark 2.2.8 and Corollary 3.2.3 it follows b1(x) = 1 and, for m ∈

{1, 2, . . . , q1 − 1}, that bm(y) = 0. Consider the word R1 = x||R1 |= x|q2 . Letn = kq1 + ( j + 1)q0 for some k ∈ {0, 1, . . . , a2 − 1} and some j ∈ {1, 2, . . . , a1}. Fork = 0,

x|n= R1|n= (0, 1, 0, 0, . . . , 0⏞ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ⏞j−1

).

By Proposition 3.2.2 and Corollary 3.2.3,

L1 = (1, 0, 0, . . . , 0⏞ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ⏞a1

)

is a right special word and thus, by Remark 2.2.8, the set of all right special wordsof length at most |L1| = a1 + 1 is

{(1, 0, 0, . . . , 0⏞ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ⏞a1

), (0, 0, . . . , 0⏞ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ⏞a1

), (0, 0, . . . , 0⏞ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ⏞a1−1

), . . . , (0, 0), (0)}.

Since there exists a unique right special word per length, it follows that bn(x) = 0.In the case that k ∈ {1, . . . , a2 − 1},

S n−|L1 |(x|n) = (0, 0, . . . , 0⏞ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ⏞a1−( j−1)

, 1, 0, 0, . . . , 0⏞ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ⏞j−1

),

where we recall that a1 − ( j − 1) ≥ 1. Since there exists a unique right specialword per length and since

|S n−|L1 |(x|n)| = |S n−|L1 |(0, 0, . . . , 0⏞ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ⏞a1−( j−1)

, 1, 0, 0, . . . , 0⏞ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ⏞j−1

)| = |L1|,

it follows that bn(x) = 0. An application of Corollary 3.2.3 completes the prooffor n ≤ |R1| = q2.

Consider the word L2 = y||L2 |= y|q3 . Let

m = lq2 + 1 + (i + 1)q1 = l|R1| + 1 + (i + 1)|L1|


for some l ∈ {0, 1, . . . , a3 − 1} and i ∈ {0, 1, . . . , a2 − 1}. By Proposition 3.2.2 wehave

S l|R1 |+1(y|m) = S l|R1 |+1(L2|m) = S (L1R0L1L1 . . .L1⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞i

) = R0R0 . . .R0⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞a1+1=q1=|L1 |

L1L1 . . .L1⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞i

and hence |S l|R1 |+1(y|m)| = (i + 1)|L1| = (i + 1)q1. By Remark 2.2.8 and Corol-lary 3.2.3,

S 1+(a2−(i+1))|L1 |(x|q2) = S 1+(a2−(i+1))q1(x|q2) = S 1+(a2−(i+1))q1(R1) = L1L1 . . .L1⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞i+1

is a right special word of length (i + 1)|L1| = (i + 1)q1. Since there is a uniqueright special word per length and since

R0R0 . . .R0⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞a1+1=q1=|L1 |

L1L1 . . .L1⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞i

≠ L1L1 . . .L1⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞i+1

it follows that bm(y) = 0. An application of Corollary 3.2.3 yields the result form ≤ |L2| = q3.

Assume there is r ∈ N+ so that the result holds for all natural numbers n < q2r

and m < q2r+1, namely,

1. bn(x) = 1 if and only if n = jq2k−1 + q2k−2 for k ∈ {1, 2, . . . , r} and j ∈{0, 1, . . . , a2k − 1}, and

2. bm(y) = 1 if and only if m = iq2l + q2l−1 for l ∈ {1, 2, . . . , r} and i ∈{0, 1, . . . , a2l+1 − 1}.

The proof of 1. and 2. for r + 1 follows in the same manner; thus below we onlyprovide the proof of 1. for r + 1. To this end consider the word

x||Rr+1 |= Rr+1 = RrLr+1Lr+1 . . .Lr+1⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞a2(r+1)

.

By way of contradiction, suppose there exists an integer n with |Rr| < n ≤ |Rr+1|, nis not of the form stated in Corollary 3.2.8 (1) and bn(x) = 1. For if not, the resultis a consequence of Corollary 3.2.3. By our hypothesis, we have,

n = |Rr| + (a2(r+1) − 1 − b)|Lr+1| + |Lr| + (a2(r+1)−1 − a)|Rr|,

where a ∈ {1, 2, . . . a2(r+1)−1} and b ∈ {0, 1, . . . , a2(r+1) − 1}. Set

v = RrLr+1Lr+1 . . .Lr+1⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞a2(r+1)−1−b

Lr RrRr . . .Rr⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞a2(r+1)−1−a

, w = RrRr . . .Rr⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞a

Lr+1Lr+1 . . .Lr+1⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞b

,

3.3. SPECTRAL METRICS ON STURMIAN SUBSHIFTS 49

so that |v| = n, |w| = Rr+1 − n, x||Rr+1= Rr+1 = vw and |S |w|(Rr+1)| = |v|. Corol-lary 3.2.3 implies S |w|(x||Rr+1) = S |w|(Rr+1) is a right special word. Since we haveassumed that bn(x) = 1 and since there exists a unique right special word perlength, Remark 2.2.8, it follows that S |w|(Rr+1) = v. If a = 1, then

S |w|(Rr+1) = S |Rr |+b|Lr+1 |(RrLr+1Lr+1 . . .Lr+1⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞a2(r+1)

) = Lr+1Lr+1 . . .Lr+1⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞a2(r+1)−b

.

This is a contradiction to the assumption bn(x) = 1; since if this were the case wewould have that S |w|(Rr+1) = v, but the first letter of v is 0 and the first letter ofS |w|(Rr+1) is 1. Hence, a ≥ 2, and so

S |w|(Rr+1)

= S a|Rr |+b|Lr+1 |(RrLr+1Lr+1 . . .Lr+1⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞a2(r+1)

)

= S (a−1)|Rr |(Lr RrRr . . .Rr⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞a2(r+1)−1

Lr+1Lr+1 . . .Lr+1⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞a2(r+1)−b−1

)

= S |Rr |−|Lr |(RrRr . . .Rr⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞a2(r+1)−1−(a−2)


)

= S (a2r−1)|Lr |+|Rr−1 |(Rr−1LrLr . . .Lr⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞a2r

RrRr . . .Rr⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞a2(r+1)−1−(a−2)−1


)

= LrRrRr . . .Rr⏞ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ⏞a2(r+1)−1−(a−2)−1


),

where we observe a2(r+1) − (a− 2)− 1 ≥ 1 and a2(r+1) − b− 1 ≥ 0. This contradictsthe assumption bn(x) = 1; since if this were the case we would have S |w|(Rr+1) = v,but the first letter of v is 0 and the first letter of S |w|(Rr+1) is 1. �

3.3 Spectral metrics on Sturmian subshiftsIn this section the spectral metric as introduced in [47] and sequential Holderregularity of metrics will be defined. Spectral metrics are also considered forspectral triples, see [46, 47] for the definition of the spectral triple used to definethe spectral metric. As the investigation in this work is solely on the combinatorialproperties of the spectral metric, a combinatorial version of the spectral metric asintroduced in [47] is given without presenting the link to spectral triples.

Definition 3.3.1 (Metric). Let X denote a Sturmian subshift of slope ξ. The metricdt : X × X → [0,∞) is defined by

dt(v,w) ≔ |v ∨ w|−t.


Definition 3.3.2 (Spectral metric). Let X denote a Sturmian subshift of slope ξ.The spectral metric dξ,t : X × X → [0,∞] is defined by

dξ,t(v,w) ≔ |v ∨ w|−t +∑

n>|v∨w|

bn(v)n−t +∑

n>|v∨w|

bn(w)n−t, (3.7)

for all v,w ∈ X. Here, for n ∈ N+ and u ∈ X, where

bn(u) =

⎧⎪⎪⎨⎪⎪⎩1 if u|n is a right special word,0 otherwise.

Let ξ = [0; a1 + 1, a2, . . . ] ∈ Θα ∩ [0, 1/2], let X denote a Sturmian subshift ofslope ξ and let x, y ∈ X denote the unique infinite words such that x||Rn |= Rn andy||Ln |= Ln, for all n ∈ N+. By Proposition 3.2.2, we have, for all n ∈ N+, that

S |Ln |(y)||Rn |+1 = Rn(0),

dt(x, S |Ln |(y)) = q−t2n,

S |Rn |(x)||Ln+1 |+1 = Ln+1(1),

dt(S |Rn |(x), y) = q−t2(n+1)−1.

(3.8)

Combining Corollary 3.2.8 and (3.7), we obtain that

dξ,t(x, S |Ln |(y)) =∞∑

k=2n

ak+1∑j=1

( jqk + qk−1 − 12Z(k − 2n)q2n−1)−t,

dξ,t(S |Rn |(x), y) =∞∑

k=2n+1

ak+1∑j=1

( jqk + qk−1 − 12Z(k − (2n + 1))q2n)−t.

(3.9)

The following result gives a necessary condition for when the spectral metric dξ,tis not bounded; complementing [47, Theorem 4.14].

Proposition 3.3.3. Let α > 1 and let X be a Sturmian subshift of slope ξ ∈ Θα.

For t ∈ (0, 1− 1/α], the spectral metric dξ,t is not a metric and for t > 1− 1/α, thespectral metric dξ,t is a metric.

Proof. For the first part of the result let ξ = [0; a1+1, a2, . . . ] belong the to [0, 1/2].Since α ≥ 1/(1 − t) > 1, 0 < Aα(ξ) = lim supn→∞ an/qα−1

n and since (qk)k∈N+ isan unbounded monotonic sequence, the sequence (an)n∈N+ is unbounded. Hence,there exists a sequence of natural numbers (mi)i∈N+ so that

0 < min{

1,Aα(ξ)

2

}< ami+1q1−α

mi≤ ami+1q1−1/(1−t)

miand ami ≥ 4.

With that for any n ∈ N+ either one of the cases stated in (3.9) equals the followingsum, which will proof that dξ,t is not a metric and for t > 1 − 1/α∞∑

k=n

ak+1∑j=1

( jqk + qk−1 − 12Z(k − n)qn−1)−t ≥

∞∑k=n

ak+1∑j=1

1( jqk + qk−1)t


≥

∞∑k=n

12tqt

k

ak+1∑j=1

1jt

≥1 − 2t−1

2t(1 − t)

∞∑i=n

a1−tmi+1q−t

mi

≥1 − 2t−1

2t(1 − t)

∞∑i=n

(ami+1q1−1/(1−t)

mi

)1−t

≥1 − 2t−1

2t(1 − t)

∞∑i=n

(min

{1,

Aα(ξ)2

})1−t

= ∞.

For the second part of the result, in [47] it has already been shown that dξ,t is apseduo metric; and thus, it remains to show that dξ,t(w, v) < ∞, for all w, v ∈ X.However, this follows directly from Proposition 3.3.8, 3.3.10 and 3.3.14. �

In context of the last proposition, it will always be assumed that υ = (n−t)n∈N+ ,where t > 1 − 1/α from here on onwards. What follows will be a comparison ofthe two metrics dt and dξ,t against each other.

Definition 3.3.4. Let d1, d2 be two metrics on X and r > 0. The metric d1 isr-Holder continuous with respect to d2 if it exists a C > 0 such that

d1(v,w)r ≤ Cd2(v,w),

for all v,w ∈ X.

It is immediate from the definition of the two metrics that dt ≤ dξ,t, but in mostcases dξ,t is not even Holder continuous to dt. Though, there is another bound fordξ,t via dt, which will be called Holder regularity and is sufficient in many caseswhere Holder continuity is not enough. In order to define it set for w ∈ X, whereX is a Sturmian subshift of slope ξ and r > 0,

ψw(r) ≔ lim supv−→

dtw

dξ,t(w, v)dt(w, v)r and ψ(r) ≔ sup{ψw(r) : w ∈ X}. (3.10)

For all r ∈ (0, 1), we will see in Proposition 3.3.13 that, by replacing limit superiorwith limit inferior in the definition of ψw(r), then ψ(r) = 0. If ψ(r) ≕ C < ∞, thenby Lebesgue’s number theorem as X is compact with respect to dt, for any coverX with open sets there is a radii R such that by dξ,t(v,w) ≤ Cdt(v,w)r for all v,wwith dt(v,w) < R a ball with respect to dξ,t is contained in at least one of the opensets of the cover of X. This let us partially understand the spectral metric via dt

and explains the term Holder used in the following definition.


Definition 3.3.5. Let r > 0 be given.

1. The metric dξ,t is sequentially r-Holder regular to dt if ψ(r) < ∞.

2. The metric dξ,t is sequentially r-Holder regular to dt if ψ(r) > 0.

3. The metric dξ,t is sequentially r-Holder regular to dt if 0 < ψ(r) < ∞.

We will also require the following weaker notion of sequential Holder regu-larity.

Definition 3.3.6. Let r > 0 be given.

1. The metric dξ,t is critically sequentially r-Holder regular to dt if ψ(r−ϵ) = 0,for all 0 < ϵ < r.

2. The metric dξ,t is critically sequentially r-Holder regular to dt if ψ(r + ϵ) =∞, for all ϵ > 0.

3. The metric dξ,t is critically sequentially r-Holder regular to dt if dξ,t is criti-cally sequentially r- and r-Holder regular to dt.

For a given r ∈ (0, 1], if the metric dξ,t is sequentially r-Holder (respectively,r-Holder) regular to dt, then dξ,t is critically sequentially r-Holder (respectively,r-Holder) regular to dt.

3.3.1 Subsequences of ψz approximandsFrom here on onwards for a Sturmian subshift X of slope ξ the sequences x, y ∈ Xwill always denote the unique infinite words with x||Rk |= Rk and y||Lk |= Lk, forall k ∈ N+. The equalities we have met in (3.9) will help to understand ψ byintroducing the following definition.

Definition 3.3.7. For n ∈ N and r > 0 set

ψx,n(r) ≔dξ,t(x, S |Ln |(y))dt(x, S |Ln |(y))r and ψy,n(r) ≔

dξ,t(S |Rn |(x), y)dt(S |Rn |(x), y)r .

Notice that lim supn→∞

ψz,n(r) ≤ ψz(r) ≤ ψ(r) for z ∈ {x, y}. The next proposition

is the first step of approximating ψz via ψz,n for z ∈ {x, y} and n ∈ N.

Proposition 3.3.8. Let α > 1 and let X denote a Sturmian subshift of slope ξ ∈[0, 1/2] and let t > 1 − 1/α.


1. (a) If t ∈ (1 − 1/α, 1) and Aα(ξ) < ∞, then

supz∈{x,y}

lim supn→∞

ψz,n(r)

⎧⎪⎪⎨⎪⎪⎩= 0 if 0 < r < α − (α − 1)/t),< ∞ if r = α − (α − 1)/t.

(b) If t ∈ (1 − 1/α, 1) and Aα(ξ) > 0, then

supz∈{x,y}

lim supn→∞

ψz,n(r)

⎧⎪⎪⎨⎪⎪⎩= ∞ if r > α − (α − 1)/t,> 0 if r = α − (α − 1)/t).

2. (a) If t = 1, Aα(ξ) < ∞ and r ∈ (0, 1), then

supz∈{x,y}

lim supn→∞

ψz,n(r) = 0,

(b) If t = 1 and if r ≥ 1, then

supz∈{x,y}

lim supn→∞

ψz,n(r) = ∞.

3. (a) If t > 1, then

supz∈{x,y}

lim supn→∞

ψz,n(r)

⎧⎪⎪⎨⎪⎪⎩= 0 if 0 < r < 1,< ∞ if r = 1.

(b) If t ∈ (1 − 1/α, 1), then

supz∈{x,y}

lim supn→∞

ψz,n(r)

⎧⎪⎪⎨⎪⎪⎩= ∞ if r > 1,> 0 if r = 1.

Remark 3.3.9. In the proof of all three parts of Proposition 3.3.8, we will use thefollowing observation. Due to qk+1 = ak+1qk + qk−1 ≥ qk + qk−1 and q0 = 1, q1 =

a1 ≥ 1, a lower bound on (qk)k∈N is given by the Fibonacci sequence ( fk)k∈N+ , thatis, f1 = 1, f2 = 1 and fk+1 = fk + fk−1. One has qk+ j ≥ f j+1qk, for all k ∈ N+ andj ∈ N by an inductive argument; qk+0 ≥ f1qk and qk+1 = qk + qk−1 ≥ f2qk and if thestatement holds for all j − 1, j − 2, then qk+ j = qk+ j−1 + qk+ j−2 ≥ f jqk + f j−1qk =

f j+1qk. Note that the j + 1 in the formula is due to fk being defined for all k ≥ 1,whether qk is given for all k ≥ 0. Setting γ ≔ (1 +

√5)/2, it is known that

fk = (γk − (−γ)−k)/√

5 and so, fk > γk/(2√

5). Thus, we have qk+ j > qkγj/(2√

5),for k ∈ N+ and j ≥ 0.


Proof of Proposition 3.3.8 1a. Since Aα(ξ) < ∞, for any c > 0 there exists aM ∈ N with ak+1q1−α

k < c, for all k > M. This with Remark 3.3.9 and the factt ∈ (1 − 1/α, 1) yields the following.

lim supm→∞

qtrm

∞∑k=m

ak+1∑j=1

1( jqk + qk−1 − 12Z(k − m)qm−1)t

≤ lim supm→∞

qtrm

∞∑k=m

1qt

k

ak+1∑j=1

1jt

≤ lim supm→∞

1 + 2t

1 − tqtr

m

∞∑k=m

a1−tk+1

qtk

= lim supm→∞

1 + 2t

1 − tqtr

m

∞∑k=m

(ak+1

qα−1k

)1−t 1

q1−α(1−t)k

≤ lim supm→∞

(1 + 2t)c1−t

1 − tqtr

m

∞∑k=m

1

q1−α(1−t)k

≤ lim supm→∞

(1 + 2t)(2√

5)1−α(1−t)c1−t

1 − tqtr

m

∞∑j=0

1

q1−α(1−t)m γ j(1−α(1−t))

= lim supm→∞

(1 + 2t)(2√

5)1−α(1−t)c1−t

(1 − t)(1 − γ−(1−α(1−t)))qtr−1+α(1−t)

m .

(3.11)

This latter value is equal to zero if 0 < r < α−(α−1)/t, and finite if r = α−(α−1)/t.This together with (3.8) and (3.9) yields that, for r ∈ (0, α − (α − 1)/t),

supz∈{x,y}

lim supn→∞

ψz,n(α − (α − 1)/t) < ∞ and supz∈{x,y}

lim supn→∞

ψz,n(r) = 0.

This completes the proof. �

Proof of Proposition 3.3.8 1b. Since Aα(ξ) > 0 and (qn)n∈N+ is an unbounded mo-notonic sequence, there exists a sequence of natural numbers (nk)k∈N+ , such thatankq

1−αnk

> Aα(ξ)/2 for all k ∈ N+. Hence, we have the following chain of inequal-ities.

lim supm→∞

qtrm

am+1∑j=1

1( jqm)t ≥ lim sup

m→∞qtr−t

m

am+1∑j=1

1jt

≥ lim supm→∞

qtr−tm

a1−tm+1 − 11 − t

≥

(Aα(ξ)

2(1 − t)

)1−t

lim supj→∞

qtr−t+(1−t)(α−1)n j


This latter term is positive and finite if r = α − (α − 1)/t and is infinite if r >α − (α − 1)/t. Combining this with (3.8) and (3.9) yields the result. �

Proof of Proposition 3.3.8 2a. Since Aα(ξ) < ∞, there exists a constant c > 1so that ak+1q1−α

k < c, for all k ∈ N+. We recall that the sequence (qk)k∈N+ isstrictly increasing and notice, for x > e1, that the function x ↦→ ln(x)/x is strictlydecreasing. Combining these observations with Remark 3.3.9 yields the followingchain of inequalities.

lim supm→∞

qrm

∞∑k=m

ak+1∑j=1

1jqk + qk−1 − 12Z(k − m)qm−1

≤ lim supm→∞

qrm

∞∑k=m

1qk

ak+1∑j=1

1j

≤ lim supm→∞

qrm

∞∑k=m

ln(ak+1) + 1qk

≤ lim supm→∞

qr−1m (ln(c) + (α − 1) ln(qm) + 1) + qr

m

∞∑k=m+1

ln(c) + (α − 1) ln(qk) + 1qk

≤ lim supm→∞

qr−1m (ln(c) + (α − 1) ln(qm) + 1) +

2√

5qr−1m

∞∑j=1

ln(c) + (α − 1) j ln(γ) − (α − 1) ln(2√

5) + 1γ j

For r ∈ (0, 1) this latter value is zero, and thus, by (3.8) and (3.9), we have

supz∈{x,y}

lim supn→∞

ψz,n(r) = 0.

This completes the proof �

Proof of Proposition 3.3.8 2b. For r ≥ 1 we have that

lim supm→∞

qrm

am+1∑j=1

1jqm≥ lim sup

m→∞qr−1

m

am+1∑j=1

1j≥ lim sup

m→∞

am+1∑j=1

1j≥ lim sup

m→∞ln(am+1).

Since Aα(ξ) > 0, the continued fraction entries of ξ are unbounded and so thislatter value is infinite. Combining this with (3.8) and (3.9) gives the requiredresult. �

Proof of Proposition 3.3.8 3a. Using Remark 3.3.9 and the assumption that t > 1,we conclude the following chain of inequalities.

lim supm→∞

qtrm

∞∑k=m

ak+1∑j=1

1( jqk + qk−1 − 12Z(k − m))t ≤ lim sup

m→∞qtr

m

∞∑k=m

1qt

k

ak+1∑j=1

1jt


≤ lim supm→∞

qtrm

∞∑k=m

t(t − 1)qt

k

≤ lim supm→∞

tt − 1

qtrm

∞∑k=m

1qt

k

≤ lim supm→∞

tt − 1

qtrm

∞∑j=0

(2√

5)t

qtmγ

jt

= lim supm→∞

t(2√

5)tqt(r−1)m

(t − 1)(1 − γ−t)

For r ∈ (0, 1) we observe that this latter value is zero and for r = 1 that it is finite.This in tandem with (3.8) and (3.9) yields that, for r ∈ (0, 1),

supz∈{x,y}

lim supn→∞

ψz,n(1) < ∞ and supz∈{x,y}

lim supn→∞

ψz,n(r) = 0.


Proof of Proposition 3.3.8 3b. Observe that

lim supm→∞

qrtm

am+1∑j=1

1( jqm)t ≥ lim sup

m→∞qt(r−1)

m

am+1∑j=1

1jt

≥ lim supm→∞

qt(r−1)m

1 − (am+1 + 1)1−t

t − 1

≥1 − 21−t

t − 1lim sup

m→∞qt(r−1)

n .

This with (3.8) and (3.9) yields, for r > 1,

supz∈{x,y}

lim supn→∞

ψz,n(1) > 0 and supz∈{x,y}

lim supn→∞

ψz,n(r) = ∞.


In the following another approach is done to derive a lower bound on ψz withregards to Aα, which will complete the groundwork to link Aα to Holder regularity


in Proposition 3.3.14. By Proposition 3.2.2, we have

x||Rn |+ j|Ln+1+1= RnLn+1 . . .Ln+1⏞ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ⏞j

(1),

y||Ln |+i|Rn |+1= Ln Rn . . .Rn⏞ˉ ˉ⏟⏟ˉ ˉ⏞i

(0),

S (a2(n+1)− j+1)|Ln+1 |(y)||Rn |+ j|Ln+1+1= RnLn+1 . . .Ln+1⏞ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ⏞j

(0),

S (a2(n+1)−1−i+1)|Rn |(x)||Ln |+i|Rn |+1= Ln Rn . . .Rn⏞ˉ ˉ⏟⏟ˉ ˉ⏞i

(1).

(3.12)

for all n ∈ N+, j ∈ {1, 2, . . . , a2(n+1)} and i ∈ {1, 2, . . . , a2(n+1)−1}. The wordsS (a2(n+1)+1)|Ln+1 |(y) and S |Ln |(y) are distinct and as we will shortly see, although theultra metric distance between these words and x are equal, the respective spectraldistances are not equal; the same holds for S (a2(n+1)−1+1)|Rn |(x) and S |Rn |(x) and theirultra metric distance, respectively their spectral distance, to y. For n ∈ N+, j ∈{1, 2, . . . , a2(n+1)} and i ∈ {1, 2, . . . , a2(n+1)−1}, we have

dt(x, S (a2(n+1)− j+1)|Ln+1 |(y)) = ( jq2(n+1)−1 + q2n)−t,

dt(S (a2(n+1)−1−i+1)|Rn |(x), y) = (iq2n + q2n−1)−t,(3.13)

and combining Corollary 3.2.8 and (3.7), we obtain that

dξ,t(x,S (a2(n+1)− j+1)|Ln+1 |(y))

=

a2(n+1)∑l≥ j

(lq2(n+1)−1 + q2n)−t+ (3.14)

∞∑k=2(n+1)

ak+1∑l=1

(lqk + qk−1 − 12Z(k − 2(n + 1))(a2(n+1) − j + 1)q2(n+1)−1)−t

dξ,t(y,S (a2(n+1)−1−i+1)|Rn |(x))

=

a2(n+1)−1∑l≥i

(lq2n + q2n−1)−t+ (3.15)

∞∑k=2(n+1)−1

ak+1∑l=1

(lqk + qk−1 − 12Z(k − (2(n + 1) − 1))(a2(n+1)−1 − i + 1)q2n)−t.

These observations lead to the upcoming definition, which will partake in the next


proposition. For n ∈ N+, j ∈ {1, 2, . . . , a2(n+1)}, i ∈ {1, 2, . . . , a2n+1} and r > 0 set

ψ( j)x,n(r) ≔

dξ,t(x, S (a2(n+1)− j+1)|Ln+1 |(y))dt(x, S (a2(n+1)− j+1)|Ln+1 |(y))r

,

ψ(i)y,n(r) ≔

dξ,t(S (a2(n+1)−1−i+1)|Rn |(x), y)dt(S (a2(n+1)−1−i+1)|Rn |(x), y)r

.

(3.16)

Again notice that lim supn→∞

{ψ( j)z,n(r) : j viable choice} ≤ ψz(r) ≤ ψ(r) for z ∈ {x, y}.

Proposition 3.3.10. Let α > 1 and let t > 1 − 1/α.

1. If Aα(ξ) < ∞, then

supz∈{x,y}

lim supn→∞

sup{ψ( j)

z,n(r) : j ∈ {1, . . . , a2(n+1)−1y(z)}

}⎧⎪⎪⎨⎪⎪⎩= 0 if 0 < r < 1 − (α − 1)/(αt),< ∞ if r = 1 − (α − 1)/(αt).

2. If Aα(ξ) > 0, then

supz∈{x,y}

lim supn→∞

sup{ψ( j)

z,n(r) : j ∈{1, . . . , a2(n+1)−1y(z)}

}⎧⎪⎪⎨⎪⎪⎩= ∞ if r > 1 − (α − 1)/(αt),> 0 if r = 1 − (α − 1)/(αt).

We divide the proof of each part of the above proposition into three cases: thefirst case when t ∈ (1 − 1/α, 1), the second case when t = 1 and the third casewhen t > 1. We will also use the following lemma and remark in the proof ofProposition 3.3.10.

Remark 3.3.11. In the upcoming Lemma 3.3.12 the term in (3.17) is related tothe term in (3.14) for m is even and to the term in (3.15) if m is odd. For, thelemma states that the latter part of the sums in (3.14) and (3.15) does not matter ifdivided by their respective terms in (3.13), when n tends to infinity.

Lemma 3.3.12. Let α > 1 and let t > 1−1/α. Let X denote a Sturmian subshift ofslope ξ ∈ [0, 1/2] where Aα(ξ) < ∞. Given r ∈ (0,min{1, α− (α−1)/t}) and givenϵ > 0, there exists M = Mt,r ∈ N+ such that for all m ≥ M and j ∈ {1, 2, . . . , am+2},

( jqm+1 + qm)tr∞∑

k=m+2

ak+1∑l=1

1(lqk + qk−1 − 12Z(k − (m + 2))(am+2 − j + 1)qm+1)t (3.17)

belongs to the open interval (0, ϵ).


Proof of Lemma 3.3.12. The lower bound follows trivial since the quantities in-volved are non-negative. Since Aα(ξ) < ∞ there exists a constant c > 1 so thatam+1 ≤ cqα−1

m , for all m ∈ N+, and hence, for all j ∈ {1, 2, . . . , am+2}, we have thefollowing chain of inequalities, where Iα ≔ (1 − 1/α, 1).

( jqm+1 + qm)tr∞∑

k=m+2

ak+1∑l=1

1(lqk + qk−1 − 12Z(k − (m + 2))(am+2 − j + 1)qm+1)t

≤

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1 + 2t

1 − tqtr

m+2

∞∑k=m+2

(ak+1qk + qk−1)1−t

qkif t ∈ Iα

3qtrm+2

∞∑k=m+2

ln(ak+1qk + qk−1)qk

if t = 1

1 + 2t

t − 1

⎛⎜⎜⎜⎜⎜⎝ ( jqm+1 + qm)1−t+tr

qm+2+ qtr

m+2

∞∑k=m+3

q1−tk

qk

⎞⎟⎟⎟⎟⎟⎠ if t > 1

≤

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1 + 2t

1 − tc1−t21−tqtr

m+2

∞∑k=m+2

1

qαt−(α−1)k

if t ∈ Iα

3qrm+2

∞∑k=m+2

ln(2c) + α ln(qk)qk

if t = 1

1 + 2t

t − 1

⎛⎜⎜⎜⎜⎜⎝( jqm+1 + qm)t(r−1) + qtrm+2

∞∑k=m+3

1qt

k

⎞⎟⎟⎟⎟⎟⎠ if t > 1

≤

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1 + 2t)c1−t21−t2√

51 − t

qtr−αt+(α−1)m+2

∞∑i=0

1γi(αt−(α−1)) if t ∈ Iα

6√

5qr−1m+2 ln(qm+2)

∞∑i=0

ln(2c) + 1 + αi ln(γ)γi if t = 1

1 + 2t

t − 1

⎛⎜⎜⎜⎜⎜⎝(qm+1 + qm)t(r−1) + 2√

5qt(r−1)m+2

∞∑i=1

1γit

⎞⎟⎟⎟⎟⎟⎠ if t > 1

In the last inequality we have used the result given in Remark 3.3.9 and thefact, for x > e1, that the function x ↦→ ln(x)/x is strictly decreasing. Sincer ∈ (0,min{1, α − (α − 1)/t}), γ > 1, t > 1 − 1/α and the sequence (qn)n∈N+ isunbounded and monotonically increasing, the result follows. �

In the proof of Proposition 3.3.10 and Proposition 3.3.13 the following defini-tion inspired by Lemma 3.3.12 will be used. Here φ corresponds to the first partof the term in (3.14) when m even and to the first part of the term in (3.15) if mis odd, where in both cases the terms are divided by their respective counterparts


given in (3.13). Given m ∈ N+, j ∈ {1, 2, . . . , am+2}, r > 0 and t > 0 set

φ(m, j, r, t) ≔ ( jqm+1 + qm)tram+2∑l= j

1(lqm+1 + qm)t . (3.18)

Proof of Proposition 3.3.10 1. To prove Proposition 3.3.10, by the terms at (3.13),(3.14), (3.15) and Lemma 3.3.12, it is sufficient to show that if Aα(ξ) < ∞, then

lim supm→∞

sup {φ(m, j, r, t) : j ∈ {1, 2, . . . , am+2}}

⎧⎪⎪⎨⎪⎪⎩= 0 if 0 < r < 1 − (α − 1)/(αt),< ∞ if r = 1 − (α − 1)/(αt),

and if Aα(ξ) > 0, then

lim supm→∞

sup {φ(m, j, r, t) : j ∈ {1, 2, . . . , am+2}}

⎧⎪⎪⎨⎪⎪⎩= ∞ if r > 1 − (α − 1)/(αt),> 0 if r = 1 − (α − 1)/(αt).

Case t ∈ (1 − 1/α, 1): Since Aα(ξ) < ∞, there exists a constant c > 0,M ∈ N sothat am+1 ≤ cqα−1

m , for all m ≥ M. With this at hand, for 0 < r ≤ 1 − (α − 1)/(αt),we may deduce the following chain of inequalities.

lim supm→∞

sup1≤ j≤am+2

φ(m, j, r, t) ≤ lim supm→∞

qrtm+2

am+2∑l=1

1(lqm+1 + qm)t

≤ lim supm→∞

1 + 2t

1 − tqrt

m+2(am+1qm+1 + qm)1−t

qm+1

≤ lim supm→∞

1 + 2t

1 − tqrt

m+2

q1−tm+2

qm+1

≤ lim supm→∞

1 + 2t

1 − t(2c)1−t(1−r)qα−αt(1−r)−1

m+1 .

(3.19)

This in tandem with the fact that 1 − (α − 1)/(αt) < α − (α − 1)/t if and onlyif t > 1 − 1/α and that (qn)n∈N+ is an unbounded monotonic sequence, yields theresult.

Case t = 1: Since Aα(ξ) < ∞, there exists a constant c > 1 so that am+1 ≤

cqα−1m , for all m ∈ N+, and since, for r > 0, the function x ↦→ xr (ln(am+2) − ln(x)),

with domain [0,∞), is maximised at x = am+2e−1/r, we have

lim supm→∞

sup1≤ j≤am+2

φ(m, j, r, t) ≤ lim supm→∞

sup1≤ j≤am+2

qr−1m + 2rqr−1

m+1 jram+2∑

l= j+1

1l

≤ lim supm→∞

sup1≤ j≤am+2

qr−1m + 2rqr−1

m+1 jr(ln(am+2) − ln( j))


≤ lim supm→∞

qr−1m +

2re−1

rqr−1

m+1arm+2

≤ lim supm→∞

qr−1m +

2re−1cr

rqαr−1

m+1 ,

for 0 < r ≤ 1 − (α − 1)/(αt) = 1/α. This in tandem with the fact that (qn)n∈N+ is amonotonic unbounded sequence, yields the result.

Case t > 1: Since Aα(ξ) < ∞, there is a constant c > 1 with am+1 ≤ cqα−1m ,

for all m ∈ N+. Further, since 0 < r ≤ 1 − (α − 1)/(αt) and since (qn)n∈N+ is anunbounded monotonic sequence, we may deduce the following chain of inequali-ties.

lim supm→∞

sup1≤ j≤am+2

φ(m, j, r, t)

≤ lim supm→∞

sup1≤ j≤am+2

qt(r−1)m + 2trqt(r−1)

m+1 jtram+2∑

l= j+1

1lt

≤ lim supm→∞

sup1≤ j≤am+2

qt(r−1)m +

2tr

t − 1qt(r−1)

m+1 jt(r−1)+1

≤

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩lim sup

m→∞qt(r−1)

m +2tr

t − 1qt(r−1)

m+1 if r ≤ 1 − 1/t

lim supm→∞

qt(r−1)m +

2tr

t − 1qt(r−1)

m+1 at(r−1)+1m+2

if 1 − 1/t < r andr ≤ 1 − (α − 1)/(αt)

≤

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩lim sup

m→∞qt(r−1)

m +2tr

t − 1qt(r−1)

m+1 if r ≤ 1 − 1/t

lim supm→∞

qt(r−1)m +

2trct(r−1)+1

t − 1qt(r−1)

m+1 q(α−1)(t(r−1)+1)m+1

if 1 − 1/t < r andr ≤ 1 − (α − 1)/(αt)

≤

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

lim supm→∞

qt(r−1)m +

2tr

t − 1qt(r−1)

m+1 if r < 1 − 1/t

lim supm→∞

qt(r−1)m +

2trct(r−1)+1

t − 1q(α−1)+αt(r−1)

m+1if 1 − 1/t < r andr ≤ 1 − (α − 1)/(αt)

lim supm→∞

qt(r−1)m +

2trct(r−1)+1

t − 1if r = 1 − (α − 1)/(αt)

This in tandem with the fact that (qn)n∈N+ is an unbounded monotonic sequence,yields the result. �

Proof of Proposition 3.3.10 2. Case t ∈ (1 − 1/α, 1): Since Aα(ξ) > 0, there isan increasing sequence of integers {nk}k∈N+ with 2ank+2 > Aα(ξ)qα−1

nk+1 > 18. Com-bining this with the fact that (qn)n∈N+ is an unbounded monotonic sequence, and


setting jm = ⌈anm+2/2⌉, we have

lim supm→∞

φ(nm, jm, r, t)

≥ lim supm→∞

(⌈anm+2/2⌉qnm+1 + qnm)trq1−t

nm+2 − (⌈anm+2/2⌉qnm+1 + qnm)1−t

(1 − t)qnm+1

≥ lim supm→∞

1 − (2/3)1−t

2tr(1 − t)

q1−t(1−r)nm+2

qnm+1

≥ lim supm→∞

(1 − (2/3)1−t)Aα(ξ)1−t(1−r)

22tr+1−t(1 − t)qα−αt(1−r)−1

nm+1 ,

where we have used (⌈anm+2/2⌉qnm+1 + qnm)tr ≥ (qtrnm+2)/(2tr) and (⌈anm+2/2⌉qnm+1 +

qnm)1−t ≤ (3/2)1−tq1−tnm+2. This in tandem with the fact that (qn)n∈N+ is an unbounded

monotonic sequence, yields the results.Case t = 1: Since Aα(ξ) > 0, there exists an increasing sequence of non-

negative integers {nk}k∈N+ so that 2ank+2 > Aα(ξ)qα−1nk+1 > 18. Setting jm = ⌈anm+2/2⌉,

we have that

lim supm→∞

φ(m, jm, r, t)

≥ lim supm→∞

( jnmqnm+1 + qnm)r

qnm+1

(ln(qnm+2) − ln( jnmqnm+1 + qnm)

)≥ lim sup

m→∞

12r

qrnm+2

qnm+1

(ln(qnm+2) − ln

(2qnm+2

3

))≥ lim sup

m→∞2−r ln(3/2)ar

nm+2qr−1nm+1

≥ lim supm→∞

2−2r ln(3/2)Aα(ξ)rqrα−1nm+1.

This in tandem with the fact that (qn)n∈N+ is an unbounded monotonic sequence,yields the results.

Case t > 1: Since Aα(ξ) > 0, there exists an increasing sequence of naturalnumbers {nk}k∈N+ so that 2ank+2 > Aα(ξ)qα−1

nk+1 > 18. Setting jm = ⌈anm+2/2⌉, wehave

lim supm→∞

φ(nm, jm, r, t)

≥ lim supm→∞

(⌈anm+2/2⌉qnm+1 + qnm)tr(⌈anm+2/2⌉qnm+1 + qnm)1−t − q1−t

nm+2

(t − 1)qnm+1

≥ lim supm→∞

(2/3)1−t − 12tr(t − 1)

q1−t(1−r)nm+2

qnm+1


≥ lim supm→∞

((2/3)1−t − 1)Aα(ξ)1−t(1−r)

22tr+1−t(1 − t)qα−αt(1−r)−1

nm+1 .

This in tandem with the fact that (qn)n∈N+ is an unbounded monotonic sequence,yields the results. �

Proposition 3.3.13. For α > 1 and r ∈ (0, 1), we have that lim infv−→

dtw

dξ,t(w, v)dt(w, v)r = 0.

Proof. By the remarks for (3.17) and φ the proof follows if both terms convergeto 0. For (3.17) this is due to Lemma 3.3.12, while for φ we notice

lim infm→∞

φ(m, am+2, r, t) = lim infm→∞

(am+2qm+1 + qm)t(r−1) = lim infm→∞

qt(r−1)m+2 = 0,

where φ(m, am+2, r, t) is as defined in (3.18). �

3.3.2 Estimates on ψw

Previously a lower bound for ψz, where z ∈ {x, y} has been investigated. To thisend the relation ψ( j)

z,n ≤ ψz were used to yield more estimates by Proposition 3.3.10.In this section an upper bound for ψw, w ∈ X will be given via ψ(k)

z,n.

Proposition 3.3.14. If Aα(ξ) < ∞, that is there exists c > 0 so that am+1 ≤ cqα−1m ,

for all m ∈ N+, then, for r > 0,

ψw(r)

≤ 2(c + 2)trsupz∈{x,y}

lim supn→∞

sup{ψ(k)

z,n(r) : k ∈ {1, . . . , a2(n+1)−1y(z)}

}∪

{ψz,n(αr)

},

where w ∈ X.

Proof. Let w = (w1,w2, . . . ) ∈ X be fixed and let n ≥ 2 denote a natural numberwith bn(w) = 1. Set kz(n) = sup{l ∈ {1, 2, . . . n} : bl(z) = 1}, where z ∈ {x, y}.(Note that kx(m(n)) ≠ ky(m(n)) as there exists a unique right special word perlength.) By definition we have bkz(n)(z) = 1, and so, by Corollary 3.2.8, there existl(n), l′(n) ∈ N+, p(n) ∈ {1, 2, . . . , a2(l(n)+1)} and p′(n) ∈ {1, 2, . . . , a2(l(n)+1)−1}, with

x|kx(n)= Rl(n)Ll(n)+1 . . .Ll(n)+1⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞p(n)

and y|ky(n)= Ll′(n) Rl′(n) . . .Rl′(n)⏞ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ⏞p′(n)

.


An application of Remark 2.2.8 and Proposition 3.2.2, yields that

inf{l ∈ N+ : bn+l(w) = 1}≥

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

|Ll(n)+1| = q2l(n)+1if wn+1 = 1, andp(n) ≠ a2(l+1),

|Rl′(n)| = q2l′(n)if wn+1 = 0, andp′(n) ≠ a2(l′(n)+1)−1,

|Ll(n)+2| = q2l(n)+3if wn+1 = 1 andp = a2(l(n)+1),

|Rl′(n)+1| = q2(l′(n)+1)if wn+1 = 0 andp′(n) = a2(l′(n)+1)−1.

(3.20)

Thus, we have that∑k≥n

bk(w)k−t ≤∑

k≥kx(n)

bk(x)k−t +∑

k≥ky(n)

bk(y)k−t. (3.21)

Let w ∈ X be fixed. Set m(0) ≔ 0, define m(n) ≔ min{k > m(n− 1) : bm(n)(w) = 1}and let (w(n))n∈N+ denote a sequence in X such that w(n)|m(n)= w|m(n) and w(n)|m(n)+1≠w|m(n)+1. Combining the above with (3.7), (3.8), (3.12) and (3.21) we conclude thefollowing.

dξ,t(w,w(n)

)≤ 2

⎛⎜⎜⎜⎜⎜⎜⎜⎝ ∑k≥kx(m(n))

bk(x)k−t +∑

k≥ky(m(n))

bk(y)k−t

⎞⎟⎟⎟⎟⎟⎟⎟⎠

≤

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2dξ,t(x, S (a2(l(m(n))+1)−p(m(n))+1)|Ll(m(n))+1 |(y))if kx(m(n)) < ky(m(n)) andp(m(n)) ≠ a2(l(m(n))+1)

2dξ,t(S (a2l′(m(n))+1−p′(m(n))+1)|Rl′(m(n)) |(x), y)if kx(m(n)) > ky(m(n)) andp′(m(n)) ≠ a2(l′(m(n))+1)−1

2dξ,t(x, S |Ll(m(n))+1 |(y))if kx(m(n)) < ky(m(n)) andp(m(n)) = a2(l(m(n))+1)

2dξ,t(S |Rl′(m(n)) |(x), y)if kx(m(n)) > ky(m(n)) andp′(m(n)) = a2(l′(m(n))+1)−1

On the other hand by (3.8), (3.13) and (3.20), for r ∈ (0, 1), we have that

dt(w,w(n))−r = (m(n))rt

≤

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2rt|Rl(m(n))Ll(m(n))+1 . . .Ll(m(n))+1⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞p(m(n))

|rt if kx(m(n)) < ky(m(n)) andp(m(n)) ≠ a2(l(m(n))+1),

2rt|Ll′(m(n)) Rl′(m(n)) . . .Rl′(m(n))⏞ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏟⏟ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ⏞p′(m(n))

|rt if kx(m(n)) > ky(m(n)) andp′(m(n)) ≠ a2(l′(m(n))+1)−1,

|Rl(m(n))+1Ll(m(n))+2|rt if kx(m(n)) < ky(m(n)) and

p(m(n)) = a2(l(m(n))+1),

|Ll′(m(n))+1Rl′(m(n))+1|rt if kx(m(n)) > ky(m(n)) and

p′(m(n)) = a2(l′(m(n))+1)−1,


≤

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2rtdt(x, S (a2(l(m(n))+1)−p(m(n))+1)|Ll(m(n))+1 |(y))−r if kx(m(n)) < ky(m(n)) andp(m(n)) ≠ a2(l(m(n))+1),

2rtdt(S (a2l′(m(n))+1−p′(m(n))+1)|Rl′(m(n)) |(x), y)−r if kx(m(n)) > ky(m(n)) andp′(m(n)) ≠ a2(l′(m(n))+1)−1,

(c + 2)trdt(x, S |Ll(m(n))+1 |(y))−αr if kx(m(n)) < ky(m(n)) andp(m(n)) = a2(l(m(n))+1),

(c + 2)trdt(S |Rl′(m(n)) |(x), y)−αr if kx(m(n)) > ky(m(n)) andp′(m(n)) = a2(l′(m(n))+1)−1.

To complete the proof we observe the following. Since dt induces the discreteproduct topology on X, any sequence in X \ {w} converging to w with respect to dt

is a subsequence of a sequence of the form (w(n))n∈N+ , and hence

ψw(r) = lim supn→∞

sup{

dξ,t(w, v)dt(w, v)r : v ∈ X, v|m(n)= w|m(n) and v|m(n)+1≠ w|m(n)+1

}.


3.3.3 Holder regularity and continued fraction expansion

10

1

Figure 3.1: Graph of ϱ8/7.

For α > 1 define the continuous function ϱα : R → Rby

ϱα(t) ≔

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩0 if t ≤ 1 − 1/α1 − (α − 1)/(αt) if 1 − 1/α < t < 1,1/α if t ≥ 1.

The function ϱα is concave on [1 − 1/α,∞) and, onthe interval (1 − 1/α, 1), it is strictly increasing, seeFigure 3.1. Also, for t ≤ 1 − 1/α one has that 1 − (α − 1)/(αt) ≤ 0.

Theorem 3.3.15. Let X be a Sturmian subshift of slope ξ, let α > 1 be given andfix t ∈ (1 − 1/α, 1).

1. The metric dξ,t is sequentially ϱα(t)-Holder regular to the metric dt if andonly if ξ ∈ Θα.

2. The metric dξ,t is sequentially ϱα(t)-Holder regular to the metric dt if andonly if ξ ∈ Θ

α.

Hence, dξ,t is sequentially ϱα(t)-Holder regular to dt if and only if ξ ∈ Θα.


Proof of Theorem 3.3.15 1. Suppose that there is a t ∈ (1 − 1/α, 1) so that themetric dξ,t is sequentially ϱα(t)-Holder regular to dt, in which case ψ(ϱα(t)) is fi-nite. If the continued fraction expansion of ξ is bounded by K, then Aα(ξ) ≤lim supn→∞ Kq1−α

n = 0 and hence ξ ∈ Θα. If the continued fraction expansion ofξ is not bounded, there is a subsequence given by anm , the m-th continued frac-tion entry of ξ, such that anm+2 ≥ 8. Since (qm)m∈N+ is a monotonically increasingunbounded sequence, we notice

Aα(ξ) = lim supm→∞

anmq1−αnm

and ⌈anm+2

2

⌉qnm+1 + qnm ≤

2(anm+2qnm+1 + qnm)3

=2qnm+2

3.

Setting r = ϱα(t) = 1 − (α − 1)/(αt), we have that

ψX(r) ≥ supz∈{x,y}

ψX,z(r)

≥ lim supm→∞

sup1≤ j≤anm+2

φ(n, j, r, t)

≥ lim supm→∞

(⌈anm+2/2⌉qnm+1 + qnm)tranm+2∑

l=⌈anm+2/2⌉

1(lqnm+1 + qnm)t

≥ lim supm→∞

(⌈anm+2/2⌉qnm+1 + qnm)trq1−t

nm+2 − (⌈anm+2/2⌉qnm+1 + qnm)1−t

(1 − t)qnm+1

≥ lim supm→∞

1 − (2/3)1−t

2tr(1 − t)

q1−t(1−r)nm+2

qnm+1

=1 − (2/3)1−t

2tr(1 − t)lim sup

m→∞

(anm+2q−t(1−r)/(1−t(1−r))

nm+1

)1−t(1−r)

=1 − (2/3)1−t

2t−(α−1)/α(1 − t)lim sup

m→∞

(anm+2q1−α

nm+1

)1/α

=1 − (2/3)1−t

2t−(α−1)/α(1 − t)Aα(ξ)1/α.

(3.22)

Hence, it follows that Aα(ξ) is finite.The reverse implication is a consequence of Propositions 3.3.8, 3.3.10 which

yield ψz ≤ ψ( j)z ≤ ∞. �

Proof of Theorem 3.3.15 2. For the forward implication, we prove the contra-po-sitive; namely that, if Aα(ξ) = 0, then ψ(1 − (α − 1)/(αt)) = 0. As before anupper bound for taken limit in the definition of ψ is given by a combination of


upper bounds for (3.17) and φ. While the first one gets arbitrary small due toLemma 3.3.12, notice that for φ the estimate given in (3.19) holds for arbitrarysmall choices of c > 0. The reverse implication follows by an identical argumentto that given in (3.22). �

Theorem 3.3.16. Let X be a Sturmian subshift of slope ξ, let α > 1 be given.

1. For t = 1, we have the following.

(a) If dξ,t is sequentially ϱα(t)-Holder regular to dt, then ξ ∈ Θα.

(b) If ξ ∈ Θα, then dξ,t is critically sequentially ϱα(t)-Holder regular to dt.

(c) If ξ ∈ Θα, then dξ,t is critically sequentially ϱα(t)-Holder regular to dt.

2. For t > 1, we have the following.

(a) If ξ ∈ Θα, then dξ,t is sequentially ϱα(t)-Holder regular to dt.

(b) If ξ ∈ Θα, then dξ,t is sequentially ϱα(t)-Holder regular to dt.

3. (a) If t ∈ (1, α/(α − 1)) and if dξ,t is sequentially ϱα(t)-Holder regular todt, then ξ ∈ Θα.

(b) If t ≥ α/(α− 1), then dξ,t is ϱα(t)-Holder continuous with respect to dt.

Proof of Theorem 3.3.16 1a. By the hypothesis we know that the metrics dξ,t anddt are not Lipschitz equivalent and so by Remark 3.3.18, the continued fractionentries of ξ are not bounded. Let anm denote the m-th continued fraction entry of ξ,such that anm+2 ≥ 8. Thus since (qm)m∈N+ is a monotonically increasing unboundedsequence, we have


anmq1−αnm

and ⌈anm+2

2

⌉qnm+1 + qnm ≤


=2qnm+2

3.

Using (3.13), (3.14), (3.15), Lemma 3.3.12 and setting jm = ⌈anm+2/2⌉ and r =ϱα(t) = 1/α, we notice that

ψX(r) ≥ supz∈{x,y}

ψX,z(r)

≥ lim supm→∞

φ(nm, jm, r, t)

≥ lim supm→∞

(⌈anm+2/2⌉qnm+1 + qnm)r ln(qnm+2) − ln(⌈anm+2/2⌉qnm+1 + qnm)qnm+1


≥ lim supm→∞

12r

qrnm+2

qnm+1(ln(qnm+2) − ln(2qnm+2/3))

≥ln(3/2)

2r lim supm→∞

a1/αnm+2q1/α−1

nm+1

≥ln(3/2)

2r lim supm→∞

(anm+2q1−αnm+1)1/α

≥ln(3/2)

2r Aα(ξ)1/α.

This yields the required results. �

Proof of Theorem 3.3.16 1b, 1c, 2a and 2b.See Propositions 3.3.8, 3.3.10 and 3.3.14. �

Proof of Theorem 3.3.16 3a. By the hypothesis we know that the metrics dξ,t anddt are not Lipschitz equivalent and so by Remark 3.3.18, the continued fractionentries of ξ are not bounded. Let anm denote the m-th continued fraction entry of ξ,such that anm+2 ≥ 8. Thus since (qm)m∈N+ is a monotonically increasing unboundedsequence, we have


anmq1−αnm

and ⌈anm+2

2

⌉qnm+1 + qnm ≤


=2qnm+2

3.

Set jm = ⌈anm+2/2⌉ and r = ϱα(t), and let φ(m, j, r, t) be as in (3.18). Using (3.13),(3.14) and (3.15) notice

ψX(r) ≥ lim supm→∞

φ(nm, jm, r, t)

≥ lim supm→∞

(⌈anm+2/2⌉qnm+1 + qnm)tr(⌈anm+2/2⌉qnm+1 + qnm)1−t − q1−t

nm+2

(t − 1)qnm+1

≥ lim supm→∞

(2/3)1−t − 12tr(t − 1)

q1−t(1−r)nm+2

qnm+1

≥(2/3)1−t − 1

2tr(t − 1)lim sup

m→∞a1−t(1−r)

nm+2 q−t(1−r)nm+1

≥(2/3)1−t − 1

2tr(t − 1)lim sup

m→∞

(anm+2q−t(1−r)/(1−t(1−r))

nm+1

)1−t(1−r)

≥(2/3)1−t − 12t/α(t − 1)

lim supm→∞

(anm+2q1−α/(α−t(α−1))

nm+1

)(α−t(α−1))/α


≥(2/3)1−t − 12t/α(t − 1)

A(α−t(α−1))(ξ)(α−t(α−1))/α,

where the last inequality holds since t ∈ (1, α/(α−1)) and hence (α− t(α−1))/α >0. Thus, Aα/(α−t(α−1))(ξ) < ∞. �

Proof of Theorem 3.3.16 3b. This is a consequence of (3.7) and the fact that, forall v,w ∈ X, with m = |v ∧ w| one has

dξ,t(v,w) ≤∞∑

n=m+1

1nt ≤

1t − 1

m−(t−1) =1

t − 1(m−t)1−1/t

≤1

t − 1(m−t)1/α =

1t − 1

dt(v,w)1/α =1

t − 1dt(v,w)ϱα(t)


Corollary 3.3.17. Let X be a Sturmian subshift of slope ξ, let α > 1 be given andfix t > 1−1/α. If ξ ∈ Θα, then the metric dξ,t is critically sequentially ϱα(t)-Holderregular to dt.

Remark 3.3.18. An analogue of Theorem 3.3.15 also holds for the case thatα = 1, see [29, 47] respectively. In this case, the sequential Holder regularityis replaced by Lipschitz equivalence. There have further been investigations formetrics induced by other sequences than (n−t)n∈N, which will be in the followingdenoted by υ = (υn)n∈N+ . In particular υ has to be a strictly decreasing null-sequence, and there exist constant c, c, such that cυn ≤ υ2n and υnm ≤ cυnυm, forall n,m ∈ N+. Indeed, in [47] the typical choice for such a sequence is suggestedto be υn = ln(n)n−t, where t > 0, and investigations of spectral metrics whenυn = n−1 are carried out in [43]. This is in line with the choice of υn = n−t made inthis work. Further, it has been shown in [47] that if the sequence υn is exponen-tially decreasing, then the metric dξ,t is Lipschitz equivalent to dt, independent ofthe Sturmian subshift. The latter part of Theorem 3.3.16 3 gives the counterpartcondition to conclude Holder continuity, independent of the Sturmian subshift.

Remark 3.3.19. In prospect of Theorem 3.2.6 the regularity conditions α-repe-titive, α-repulsive and α-finite let us deduce when dξ,t is (critically) sequentiallyϱα(t)-Holder regular to dt for (t > 1 − 1/α) or t ∈ (1 − 1/α, 1) respectively. Theconverse direction is only viable if dξ,t is sequentially ϱα(t)-Holder regular to dt

for t ∈ (1 − 1/α, 1), which let us conclude that a Sturmian subshift is α-repetitive,α-repulsive and α-finite.

Remark 3.3.20. With a look at the calculations done in this chapter it seemsreasonable to conjecture that the sequential Holder regularity in Theorems 3.3.15


and 3.3.16 cannot be strengthened to Holder equivalence. A clear indication forthat is given in Propositions 3.3.8 and 3.3.10. Also Theorems 3.3.15 and 3.3.16should still hold for υn = ℓ(n)n−t, where ℓ is a slowly varying function as it ismentioned in Remark 3.3.18 for α = 1. For further details on slowly varyingfunctions a viable source is given by [15].

3.4 Hausdorff dimension of ΘαDefinition 3.4.1. Let ψ : R → R be a strictly positive monotonically decreasingfunction, the set

Jψ ≔

{x ∈ [0, 1] :

x −

pq

≤ ψ(q) for infinitely many p, q ∈ N+

}is called the ψ-Jarnık set. When ψ(y) = cy−β, where β > 2 and c > 0, we denotethe set Jψ by Jc

β and define

Exact(β) ≔ J1β \

⋃n≥2, n∈N+

Jn/(n+1)β

to be the set of real numbers that are approximable to rational numbers p/q toorder qβ but no better.

The theory of ψ-Jarnık sets has been further studied in [21, 22] and some ofthe results obtained there will be used in the following.

Theorem 3.4.2. For β > 2 and c > 0, we have

dimH (Jcβ) = dimH (Exact(β)) = 2/β,

where dimH denotes the Hausdorff dimension.

Proof. Let Exact(β, c) ≔ Jcβ \

⋃n≥2, n∈N+ J

n/(n+1)cβ . In [22, Thm. B] it has been

shown that dimH (Exact(β, c)) = dimH (Jcβ) = 2/λ, where we have for the constant

λ ≔ lim infx→∞ log(ψ(x)−1)/ log(x) = lim infx→∞ log(c−1xβ)/ log(x) = β. Hencethe Hausdorff dimension of dimH (Jc′

β ) is the same for all c′ > 0, which proofs thetheorem. �

Theorem 3.4.3. For α > 1 we have that dimH (Θα) = dimH (Θα) = 2/(α + 1)

and m(Θα) = 1, where dimH denotes the Hausdorff dimension and m denotes theLebesgue measure on R.

3.4. HAUSDORFF DIMENSION OF Θα 71

Proof of Theorem 3.4.3. To obtain that dimH (Θα) = 2/(α + 1), we show that Θαis contained in a countable union of ψ-Jarnık sets each with the same Hausdorffdimension, namely 2/(α + 1). For ξ = [0; a1, a2, . . . ], it is known that,

1(an+1 + 2)q2

n≤

1qn(qn + qn+1)

=

pn + pn+1

qn + qn+1−

pn

qn

≤

ξ −

pn

qn

≤

pn

qn−

pn+1

qn+1

≤

1an+1q2

n,

(3.23)

see for instance [52]. Also, considering approximations (pn(x)/qn(x))n∈N+ , of anirrational number x = [0; a1, a2, . . . ] ∈ [0, 1], we have that J1/c

α+1 contains the set{x = [0; a1, . . . ] ∈ [0, 1] :

x −

pn(x)qn(x)

≤ c−1qn(x)−α−1 for infinitely many n ∈ N+

},

for further details see [52]. Thus, if lim supn→∞

an+1q1−αn ≥ c, for some given c > 0,

the lower bound given in (3.23) can be used to yield ξ ∈ J1/cα+1. Therefore,

Θα ⊆ Θα ⊆ {ξ ∈ [0, 1] : Aα(ξ) > 0} ⊆⋃n∈N+

Jnα+1,

and so, by monotonicity and countable stability of the Hausdorff dimension (seefor instance [32]) and Theorem 3.4.2, we have that

dimH (Θα) ≤ dimH (Θα) ≤ 2/(α + 1). (3.24)

To prove that 2/(α+1) is a lower bound for dimH (Θα) and dimH (Θα) we first show

Exact(α + 1) is a subset of Θ(α) and Θ(α) and hence a subset of Θ(α). By [52,Theorem 15] every best (reduced) rational approximation (of the first kind) p/q toξ = [0; a1, a2, . . .], namely |ξ − p′/q′| > |ξ − p/q|, for all p′, q′ ∈ N+ with q′ < q,is necessarily of the form p(m)/q(m) = [0; a1, a2, . . . , an−1,m], for some n ∈ N+ and1 ≤ m ≤ an. In fact, an/2 ≤ m ≤ an, since if m < an/2, then by (3.23),

ξ − p(m)

q(m)

−

ξ −

pn−1

qn−1

≥

pn−1

qn−1−

p(m)

q(m)

− 2

ξ −

pn−1

qn−1

≥

pn−1

qn−1−

p(m)

q(m)

− 2

qnqn−1

=1

q(m)qn−1−

2qnqn−1

≥(an − 2m)qn−1 − qn−2

q(m)qnqn−1> 0.


Hence, p(m)/q(m) is not a best approximation (of the first kind). From this, weconclude 1/2 ≤ q(m)/qn ≤ 1, for an/2 ≤ m ≤ an. Hence, for ever reduced fractionp/q with |ξ − p/q| ≤ q−1−α we may assume without loss of generality that p/q is abest approximation (of the first kind) and hence we find n ∈ N+ such that

ξ −pn

qn

≤

ξ −

pq

≤ q−(α+1) ≤ 2α+1q−(α+1)

n .

Using the lower bound in (3.23) gives, for every ξ ∈ Exact(α+1) a lower bound onlim sup an+1q1−α

n ≥ 2−(α+1) and thus that Exact(α+ 1) ⊂ Θ(α). Further, assume that|ξ − p/q| > dq−(α+1) for some d < 1 and all but finitely many rationals p/q. Thistogether with the upper bound in (3.23) yields that lim sup an+1q1−α

n ≤ d−1. In thisway we have verified that Exact(α + 1) ⊂ Θ(α). The statement on the Hausdorffdimension of Θ

αand Θα now follows from an application of Theorem 3.4.2, the

monotonicity of the Hausdorff dimension (see for instance [32]) and (3.24).To complete the proof, we show that m(Θα) = 1. Notice, if ξ ∈ [0, 1] \ J1

α+1,using the upper bound given in (3.23), we have that an+1q1−α

n < 1, for all butfinitely many n ∈ N+, and thus, Aα(ξ) < 1. In particular, we have Θα ⊇ [0, 1] \J1α+1. This with Theorem 3.4.2 yields m(Θα) ≥ m([0, 1] \ J1

α+1) = 1. �

Chapter 4

Measure theory

Most of the measure theory which will find use in the following chapters is cov-ered here. This includes complex-valued measures, some Riesz representationtheorems and the Radon-Nikodym derivative. For measures as functionals on thespace of continuous functions with compact support see Section 5.2.1 and Ap-pendix C.2.

A (non-negative) measure is a σ-additive, non-negative map from a σ-algebrainto [0,∞]. It is further wise to give the empty set measure zero. If the imagespace is a subset of [0,∞), the measure is called finite or bounded. One mayas well consider C instead of [0,∞) and require its variation, Definition 4.1.1, tobe a non-negative finite measure. Such measures will be called (complex-valued)measures. The term non-negative will often be used to emphasise a mapping to[0,∞], wherever it is deemed to be of importance. As we make use of conjugatesand canonical constructions such as dual spaces, all considered σ-algebras will beBorel-σ-algebras on a locally compact Hausdorff space X.

4.1 MeasurabilityFor any topological space (X,T ), the space (X,B), where B = σ(T ) is calleda measurable space and for any measure µ on B the triple (X,B, µ) is calledmeasure space. To see that the following definition is well-defined one can check[77, Thm. 6.19] or [86, A.4].

Definition 4.1.1. Let (X,B) be a measurable space and µ : B → C be aσ-additivemap. µ is then called a complex-valued measure. The (total) variation of µ is givenby

|µ|(A) ≔ sup

⎧⎪⎪⎨⎪⎪⎩∑i∈I

|µ(Ai)| : (Ai)i∈I ∈ BI is a finite disjoint partition of A

⎫⎪⎪⎬⎪⎪⎭ ,73

74 CHAPTER 4. MEASURE THEORY

for all A ∈ B and is a non-negative measure. We further set ∥µ∥ ≔ |µ|(X).

For µ being a complex-valued measure ∥µ∥ < ∞, [77, Thm. 6.4] and one saysthat µ is of bounded variation. In the case of non-negative measures, set |µ| ≔ µ.The space X is called locally finite, if for every x ∈ X there is an open set U ∋ xsuch that µ(U) < ∞. A complex-valued measure µ is said to be locally finite, if |µ|is locally finite. Specifically

Definition 4.1.2. Let (X,B, µ) be a measure space and µ be either a non-negativeor complex-valued measure.

• The measure µ is called a Borel measure if µ is locally finite.

• The measure µ is inner regular if |µ|(A) = sup{|µ|(K) : K ⊆ A,K compact}and outer regular if |µ|(A) = sup{|µ|(U) : U ⊇ A,U open} for all A ∈ B.

• The measure µ is called a Radon measure if it is an inner regular Borelmeasure. Hence it is inner regular and locally finite.

• If µ is inner regular and outer regular, it is called regular.

• The measure µ is moderate, if it is the union of a sequence of open sets onwhich µ is finite.

• M (X) denotes the space of all complex-valued regular Borel measures onX, whileM(X) denotes the space of all non-negative regular Borel measureson X. Note that an important difference lies in the finiteness of the totalvariation, which makes M (X) into a Banach space.

For every Borel measure µ an immediate consequence is |µ|(K) < ∞ for allcompact sets K ⊆ X. More details on complex-valued measures can be found in[86, A.4], another source is [77, ch. 6], who dedicates a whole chapter. A goodsource for non-negative measures is [31].

4.2 Non-negative measuresIn this section some properties of infinite non-negative measures are listed. Fromnow on we additionally assume the topological space (X,T ) to be second count-able, that is if there exists a sequence of open sets (Vi)i∈N ∈ T

N such that everyopen set is the union of a subsequence. Note that a second countable space is alsoseparable, as one can choose xi ∈ Vi for all i ∈ N, which is a sequence such thatevery open set contains at least one xi for some i ∈ N and therefore (xi)i∈N is densein X.

4.3. COMPLEX-VALUED MEASURES 75

Theorem 4.2.1 (Riesz I). Let X be a second countable locally compact Haus-dorff space and I : Cc(X) → R+ be a non-negative linear functional. There existsexactly one Borel measure µ : B → [0,∞] such that

I( f ) =∫

Xf dµ

for all f ∈ Cc(X). This measure is regular and moderate.

Proof. [31, VIII Kor. 2.6], [31, VIII, Kor. 1.12]. �

Note that within this setting X is even σ-finite with regards to Borel measures.That Borel measures are regular and moderate follows also by Ulams theorem forPolish spaces, [31, VIII, Satz 1.16].

Definition 4.2.2. A separable space X is called Polish, if there exists a metric don X, such that d generates the topology on X and such that (X, d) is a completemetric space, namely, every Cauchy sequence converges to an element of X.

Typical examples of such spaces are Z,R,C with the canonical choices ofmetrics. Others are Σ∗ and ΣN by taking cylinder sets as a base of the topology,which induces an ultra-metric, see Section 2.2.

Theorem 4.2.3 (Portmanteau). For probability measures µn, µ ∈ M1(X), n ∈ N,the following are equivalent

1. v-limn→∞ µn = µ.

2. lim supn→∞ µn(A) ≤ µ(A) for all closed sets A.

3. For any function f : X → C with its set of discontinuities U f satisfyingµ(U f ) = 0 is holds limn→∞

∫f dµn =

∫f dµ.

Proof. The statements are given in [55, Satz 17.8]. �

4.3 Complex-valued measuresThroughout this work all complex-valued measures will be finite if not explicitlystated otherwise. The exceptions in this work consist of measures on Z, which inreturn are regular moderate Borel-measures, as Z will always carry the discretetopology and the measures considered in Chapter 7 and Appendix C.2. Like theprevious section we use a Riesz representation theorem to identify a measure witha functional on C0. We start by presenting a connection to non-negative continu-ous linear functionals.


Theorem 4.3.1. Let X be a locally compact Hausdorff space and I : C′0(X) → Cbe a continuous linear functional, i.e. I ∈ C′0(X). Then there are non-negativelinear functionals J+, J−, L+, L− ∈ C′0(X,C) such that

I = J+ − J− + i(L+ − L−).

This decomposition is unique in the sense if there are non-negative linear function-als P+, P−,Q+,Q− ∈ C′0(X,C) with I = P+−P−+i(Q+−Q−), then P+−J+ = P−−J−

and Q+ − L+ = Q− − L−.

Proof. [31, VIII, Satz 2.25] �

For complex-valued measures a similar version of the Riesz representationtheorem holds and we note that two measures µ, ν are said to be mutually singular,if there is an A ⊆ X such that µ(A) = ν(Ac) = 0 and write µ ⊥ ν.

Theorem 4.3.2 (Riesz II). Let X be a locally compact Hausdorff space and I fromC0(X,C) to C be a continuous linear functional with respect to ∥ · ∥L and denoteby M (X) the space of complex-valued regular Borel measures. The mapping

Φ : M (B)→ C′0(X,C), Φ(ν)( f ) ≔∫

Xf dν, f ∈ C0(X,C)

is a norm preserving isomorphism, hence ∥ν∥ = ∥Φ(ν)∥L . Especially ν has aunique decomposition, called Jordan-decomposition, into non-negative finite reg-ular Borel measures ρ+, ρ−, ϱ+, ϱ− ∈ M(X) which are pairwise mutually singularsuch that ν = ρ+−ρ−+i(ϱ+−ϱ−). Moreover ρ+−ρ− = J+−J− and ϱ+−ϱ− = L+−L−,for every choice of J+, J−, L+, L− according to Theorem 4.3.1.

Proof. [31, VIII, Satz 2.26]. For uniqueness of the Jordan decomposition checkSatz 1.12 in Kapitel VII, as well as Definition 2.21 and Folgerung 2.22(iii) ofKapitel VIII in [31]. An alternate proof can be found in [77, Thm. 6.19]. �

In considering C0 instead of Cc the Riesz representation theorem leads to theBanach space M (X), which presents more structure thanM(X) which correspondsto the first Riesz representation theorem, 4.2.1.

4.4 Decomposition of a measureThe discussed Riesz representation theorems identify operators with measures.On the other hand, every Borel measure µ ∈ M(X) ∪M (X) defines a continuouslinear functional from Cc(X,C) to C, by f ↦→

∫X

f (x) dµ(x), where X is a locallycompact Hausdorff space. When we have identified a measure µ we can extendthe space Cc(X,C) to L1

µ(X).

4.4. DECOMPOSITION OF A MEASURE 77

Definition 4.4.1. Let (X,B, µ) be a measure space. For any f ∈ L1µ(X) we set

⟨µ, f ⟩ ≔∫

Xf (x) dµ(x).

Another point of interest is which parts of a function are actually recognisedby a measure. This is for a measure µ on B, its support is given by

supp(µ) ≔(⋃{U open : |µ|(U) = 0}

)c.

One has µ(X\ supp(µ)) = 0 and |µ|(U) > 0 for all open sets U ⊆ supp(µ), [31, p.343, VIII Lem. 2.15]. In particular for two mutually singular measure µ, ν onehas supp(µ) ∩ supp(ν) = ∅. For two measures µ, ν on B, we say ν is absolutelycontinuous with respect to µ, in symbolds ν ≪ µ, if for all A ∈ B with µ(A) = 0 itfollows ν(A) = 0. They are said to be equivalent, in symbols µ ∼ ν, if µ ≪ ν andν ≪ µ.

Theorem 4.4.2. For any complex-valued measure µ ∈M (X) and non-negativeσ-finite measure ν ∈ M(X), there exists a unique decomposition into µa, µs ∈M (X)such that

1. Lebesgue decomposition:

µ = µa + µs, µa ≪ ν, µs ⊥ ν.

2. Radon-Nikodym: There exists a unique h ∈ L1µ(X) such that for all A ∈ B

µa(A) =∫

Ah dν.

The function h is called the Radon-Nikodym derivative and is also denotedby ∂µa/∂ν.

Proof. [77, Thm. 6.9 and 6.10 Extensions of Theorem 6.9]. �

Remark 4.4.3. A more general version for not necessarily finite signed measuresis presented in [31, Ch. VII] including a discussion for complex-valued measuresin the form of exercises. The subtlety can be seen in the definition of signedmeasures [31, VII, Def. 1.1(ii)], which requires the measure to be bounded eitherbounded from below or above. A downside of this generalisation is that the spaceof such measures is not closed under addition. For complex-valued measures, amore general treatment is generally achieved by considering the dual space C′c(X).This method is e.g. described in the works of Bourbaki ([19, 16, 17, 18]) and [81,p. 53ff].


As we have seen, the notion of absolute continuity presents nice criterion tocategorise measures.

Definition 4.4.4. If for every x ∈ X the intersection done with some sequences ofopen neighbourhoods of x are equal to {x}, countable collections of points are inB. This is for example the case if X carries a metric d, as

⋂n∈N B(x, 1/n) = {x}, in

any other case there would be a y ≠ x with d(x, y) = 0, hence a contradiction. Ameasure µ is discrete or (purely) atomic, if µ ∼ δC for some countable set C ∈ B,where δC ≔

∑c∈C δc and are referred to as Dirac comb. A measure µ is called

atom-less, if for all x ∈ X one has µ ≪ δx. Take as a side-note that uncountablediscrete measures will not be considered in this work. By the Radon-Nikodymtheorem, for µ ∈M (X), there is a unique decomposition

µ = µd + µc, µd, µc ∈M (X)

where µd ∼ δC for some countable set C ⊆ X and µc is atom-less. In particular µd

and µc are then mutually singular. If µd = 0, then it holds that µ = µc and the mea-sure is called absolutely continuous. µd and µc are also called the discrete part andthe continuous part of µ, which is often done if both are non-trivial. Other veryinteresting measures are the Haar measures and therefore the Lebesgue-measurem on the appropriate spaces. The Lebesgue decomposition of µc with respect to m,such that (µc)a ≪ m is therefore interesting and (µc)a is called the Lebesgue partof µ. If µ ≪ m or µ ∼ m we say µ is absolutely continuous or equivalent to theLebesgue measure, as already covered by the primal definitions and is only men-tioned here to emphasise the huge amount of interest attributed to these specialcases. With respect to m we set µac ≔ (µc)a and µas ≔ (µc)s, such that

µ = µd + µac + µsc, and µd ⊥ µac, µd ⊥ µas, µac ⊥ µas.

Chapter 5

Fourier transformation andBochner’s theorem

In this chapter the neccessary theory will be presented to introduce spectral theoryof dynamical systems in Chapter 6. Many important theorems might be amiss ornot stated in all of their generality. Especially in the latter part of this work wewill consider subshifts, and therefore Z and [0, 1) are almost enough to work with.Even though some examples also involve R and Zq and it would not take muchadditional effort to present the framework in a more general form, it will not beconsidered here, as it is already part of many textbooks, e.g. [60, 79, 24]. Connec-tions are instead pointed out when neccessary. A short discussion for measureson locally compact abelian groups based on [3] is given in Appendix C.2. Recentworks about the field of harmonic analysis related to this are [59, 84, 42].

5.1 Fourier transform of functions

The theory of Fourier transformation uses the idea of dual representation of thespace of interest and understanding the dual helps in understanding the originalspace. This field involves many different branches, whose interplay lead to thetheory of abstract harmonic analysis. When taking the fourier transform of afunction it will always live on either Z, [0, 1),R or Zq respectively, where q ∈ N+.Each of these spaces admits Haar measures and we choose the counting measureδZ for Z, the Lebesgue measure m transported to [0, 1) for [0, 1), the Lebesguemeasure m for R and the counting measure δZq for Zq for all q ∈ N+. For easeof notation we consider (G,Γ) ∈ {(Z, [0, 1)), ([0, 1),Z), (R,R), (Zq,Zq)}, whereq ∈ N+, their corresponding Haar measure spaces are (Z,B, δZ), ([0, 1),B,m),(R,B,m), (Zq,B, δZq) respectively and we assign the measures mG, mΓ in accor-dance with that.

79

80 CHAPTER 5. FOURIER TRANSFORMATION. . .

Definition 5.1.1. The Fourier transformˆ· : L1mG

(G)→ C0(Γ) given by

ˆf (y) ≔∫

Gf (x)e−2πiyx dmG(x), y ∈ Γ

is a bounded linear operator and the inverse Fourier transform∨ : L1mΓ(Γ)→ C0(G)

f ∨(x) ≔∫Γ

f (y)e2πixy dmΓ(y), x ∈ G

is a bounded linear operator.

If f ∈ L2mG

(G), the Fourier transform can be considered as an operator on aHilbert space given by ˆf (x) = ⟨ f , ex⟩ =

∫f ex dmG for all x ∈ Γ, where ex : y ↦→

e2πixy. It is notable that {e2πixn : n ∈ Z} forms an orthonormal base of L2m([0, 1)),

[86, p. 237].

5.2 Bochner’s TheoremWe use the same choices for the measure spaces on G,Γ as in Section 5.1. Othernotions that we use throughout this chapter include ˜f (x) ≔ f (−x) for functionsf : G → C and positive definiteness.

Definition 5.2.1. A function f : G → C is called positive definite, if for all N ∈ N,c = (ci) ∈ CN and x = (xi) ∈ GN holds that∑

1≤i, j≤N

cic j f (xi − x j) ≥ 0.

The inequality in the sum is equivalent to ( f (xi−x j))1≤i, j≤N being a positive definitematrix. We denote by P(G) the space of all positive definite functions on G and byCP(G) the space of all continuous positive definite functions on G.

Three immediate consequences for positive definite functions f are f (0) ≥ 0,which can be seen by choosing N = 1, c1 = 1. For N = 2, c1 = c2 = 1, x1 = x ∈G, x2 = 0 one has 0 ≤ 2 f (0) + f (x) + f (−x), which is a real number and hencef (x) = f (−x) = ˜f (−x). For the third property let N = 2, c1 ∈ C, x1 = x ∈ G, c2 =

f (x), x2 = 0, then 0 ≤ |c1|2 f (0)+ c1| f (x)|2 + c1| f (x)|2 + | f (x)|2 f (0). If one assumes

f (0) = 0 it follows f (x) = 0. If f (0) ≠ 0, set c1 = − f (0), then the inequality canbe rearranged to | f (x)|2 ≤ f (0). Hence every positive definite function is boundedand attains its maximum at 0. The following theorem has a central role in thiswork and therefore a proof is added to it.

5.2. BOCHNER’S THEOREM 81

Theorem 5.2.2 (Bochner). For any continuous positive definite function f ∈CP(G), there exists a unique bounded non-negative measure µ ∈ M+(Γ) suchthat

f (y) =∫Γ

e2πixy dµ(x)

for all y ∈ G.

Definition 5.2.3. Every measure and/or function obtained by using Bochner’s the-orem will be called Bochner transform. That is µ is the Bochner transform of fand f is the Bochner transform of µ, where µ and f are given as in Theorem 5.2.2.

The following proof of Theorem 5.2.2 is from [79, ch. 1.4]. As it involvesGelfand representation it would not be natural to restrict on our special choices forG. In fact the only case where a difference can be spotted is in the representationof the character product (x, γ) = e2πixγ.

Proof. Let A(Γ) ≔ {ˆf : f ∈ L1µ(G)}. One main idea of the proof is that any

functional F : A(Γ) → C with ∥F∥A(Γ) ≤ 1 can be extended to a functional onC0(Γ), by [79, Thm. 1.2.4]. This let us apply the Riesz representation theorem(see [79, App. E] and [86, II.6] for additional information), to find a µ ∈ M (Γ)such that for all ψ ∈ C0(Γ)

F(ψ) =∫Γ

ψ dµ

and ∥F∥ = ∥µ∥. The other main idea is in the construction of F itself from acontinuous positive definite function f : G → C. First define T : L1

m(G) → C byf ↦→

∫Gϕ f dm, where m denotes a fixed Haar measure on G. This is well-defined

as for positive definite functions f (0) = max{| f (x)| : x ∈ G}. With that, we alsoassume without loss of generality that f (0) = 1 and hence ∥T∥L1

m≤ 1. Set

[ϕ, ψ] ≔ T (ϕ ∗ ˜ψ) =∫

G

∫Gϕ(x)ψ(x − y) dm(x) f (y) dm(y)

=

∫G

∫Gϕ(x)ψ(y) f (x − y) dm(x) dm(y),

where ϕ, ψ ∈ L1m(G). This gives an inner product on L1

m(G). Due to ϕ ↦→ [ϕ, ψ]being linear, [ϕ, ψ] = [ψ, ϕ] and [ϕ, ϕ] ≥ 0. To show this, we observe that forany ϕ ∈ Cc(G), supp(ϕ) ≕ K the function (x, y) ↦→ ϕ(x)ϕ(y) f (x − y) is uniformlycontinuous and hence for any ε > 0, there exists an N ∈ N such that∫

G

∫Gϕ(x)ϕ(y) f (x − y) dm(x) dm(y)

−∑

0≤i, j≤N

ϕ(xi)ϕ(x j) f (xi − x j)m(Ei)m(E j)

< ε, (5.1)


for a partition (Ei)Ni=0 of K, with xi ∈ Ei. Since f is positive definite the sum in

(5.1) is always non-negative and hence is the integral. Using that Cc(G) is densein L1

m(G) we deduce [ϕ, ϕ] ≥ 0. This gives us the Cauchy-Schwarz inequality|[ϕ, ψ]|2 ≤ [ϕ, ϕ][ψ, ψ], [1, Lem. 0.2]. For a fixed ε > 0 set ψ = (1/m(V))1V ,where V ∋ 0 is symmetric such that | f (x) − f (y)| < ε for any x, y ∈ G with(x − y) ∈ V , which is possible by the uniform continuity of f , [79, 1.4.1].

|[ϕ, ψ] − T (ϕ)| =∫

G

∫Gϕ(x)ψ(y) f (x − y) dm(y) dm(x) −

∫Gϕ f dm

=

∫Gϕ(x)

1m(V)

∫V

f (x − y) − f (x) dm(y) dm(x)

≤

∫G|ϕ| dm ε

1m(V)

m(V).

From f (0) = 1 we see,

|[ψ, ψ] − 1| =∫

V

1m(V)

∫V

1m(V)

f (x − y) dm(y) dm(x) − 1

=

1

m(V)2

∫V

∫V

f (x − y) − f (0) dm(y) dm(x)≤ ε

m(V)2

m(V)2 .

Hence by the Schwarz inequality and C ≔∫|ϕ| dm

|T (ϕ)|2 ≤ (|[ϕ, ψ]| +Cε)2 ≤ ([ϕ, ϕ][ψ, ψ] +Cε)2 ≤ ([ϕ, ϕ](1 + ε) +Cε)2 .

This holds for all ε > 0, hence

|T (ϕ)| ≤ [ϕ, ϕ] = T (ϕ ∗ ˜ϕ). (5.2)

What follows is an application of the Gelfand representation theorem, which letsus derive that the spectral radius is given by limn→∞ ∥ϕ

n∥1/n1 = ∥ˆϕ∥∞ for any ϕ of

the Banach algebra (L1m(G),+, ∗) over C, for information see [86, Def. IX.1.2,

Thm. IX.2.8], [79, D6,Thm. 1.2.2]. Set h ≔ ϕ ∗˜ϕ and hn ≔ hn−1 ∗ h for all n ≥ 2.Then a consecutive application of (5.2) yields |T (ϕ)|2 ≤ T (h) ≤ T (h2)2−1

≤ . . . ≤T (h2n

)2−n. Further |T (h2n

)2−n| ≤ ∥h2n

∥2−n

1 for all n ∈ N and limn→∞ ∥h2n∥2−n

1 = ∥ˆh∥∞ =

∥ˆϕ∥2∞. This gives |T (ϕ)| ≤ ∥ˆϕ∥∞, which defines a bounded linear operator on A(Γ)that is extended onto C0(Γ) as mentioned at the beginning of this proof. Thisoperator will be denoted by F and is well-defined as ˆϕ = ˆψ implies F(ϕ) = F(ψ),by |T (ϕ)| ≤ ∥ˆϕ∥∞ for all ϕ ∈ L1

m(G). Together with the Riesz representationtheorem mentioned at the beginning we can deduce∫

Gϕ(x) f (x) dm(x) = F(ϕ) =

∫Γ

ˆϕ(−γ) dµ(γ) =∫

Gϕ(x)

∫Γ

(x, γ) dµ(γ) dm(x).


Hence for a.e. x ∈ G we have f (x) =∫

(x, γ) dµ(γ) and continuity of both termsyields equality for all x ∈ G. Finally, note 1 = f (0) =

∫Γ(0, γ) dµ(γ) = µ(Γ) ≤

∥µ∥ = ∥F∥ ≤ 1, which shows that µ is a non-negative measure and unique by[79, 1.3.6]. On the contrary for any non-negative µ ∈ M+(Γ) we have that thecontinuous function x ↦→

∫Γ(x, γ) dµ(γ) is positive definite, as∑

0≤i, j≤N

cic j

∫Γ

(xi − x j, γ) dµ(γ) =∑

0≤i, j≤N

cic j

∫Γ

(xi, γ) dµ(γ)∫Γ

(x j, γ) dµ(γ)

=

∑0≤i≤N

ci

∫Γ

(xi, γ) dµ(γ)

2 ≥ 0,

which completes the proof. �

Remark 5.2.4. The Fourier transform in terms of Bochner’s theorem is a continu-ous linear operator, see [79, Ch. D, D.5, Thm.], even a homeomorphism, see [13,Thm. 3.13]. It can be extended to the span of all continuous positive definite func-tions SCP(G) and is then a map into M (Γ). By using the Jordan decompositionfor a measure in M (Γ), see Theorems 4.3.1 and 4.3.2 or see [77, Ch. 1.5], onecan see that the mapping is still a bijection and thus a homeomorphism. It is no-table to remark that SCP(G) is often called B(G) the space of all Fourier-Stieltjestransforms.

For a function f ∈ LmG (G)1 ∩ SCP(G), its Bochner transform ˆf ∈ L1mΓ(Γ) and

f (y) =∫Γ

ˆf (x)e2πixy dmΓ(x).

This result can be found in [77, Ch. 1.5, Theorem], while using that the consideredHaar-measures are exactly the duals of each other. In particular this gives theinversion formula f = (ˆf )∨. A question one might ask at this point is whichfunctions are positive definite. A popular construction is by the convolution f ∗˜f ∈ CP(G) for f ∈ L2

mG(G), [77, Exl. 1.4.2]. The proof is a straightforward

calculation that uses∑

i, j cic j f (xi − x) f (x j − x) = |∑

i ci f (xi − x)|2. Next we willshow f ∗ g ∈ SCP(G) for f , g ∈ L2

mG(G) or Cc(G). This is done by the following

linear combination of continuous positive definite functions

( f + g) ∗ ( ˜f + g) − ( f − g) ∗ ( ˜f − g) + i( f + ig) ∗ ( ˜f + ig) − i( f − ig) ∗ ( ˜f − ig)

=2 f ∗˜g + 2g ∗ ˜f + 2i( f ∗˜(ig) + ig ∗ ˜f )

=2( f ∗˜g + g ∗ ˜f + i(−i) f ∗˜g + i2g ∗ ˜f )=4 f ∗˜g, (5.3)

where we used properties of the convolution mentioned in Theorem C.1.1. Finally,by noting that ∼ is an involution, we have f ∗ g ∈ SCP(G).


5.2.1 The IntegersThe Integers are a remarkable space in context of Bochner’s theorem for manyreasons. One is that all functions f : Z → C are continuous, as Z is a discreteset, in particular P(Z) = CP(Z). More so, there is a one-to-one correspondencebetween the space C(Z) and the space of continuous linear functionals C′c(Z) givenby

C(Z) ∋ f ←→∑z∈Z

f (z) δz ∈ C′c(Z). (5.4)

For that reason we will often not distinguish between functionals and functions onZ. For cases where a distinction is deemed helpful the mapping will be denotedby f ↦→ f δZ and f δZ ↦→ f in terms of (5.4). The embedding of SCP(Z) in C′c(Z),in terms of (5.4), will be denoted by SFP(Z). This might seem important in termsof Fourier transformation, regarding the upcoming definition, but the follow-uptheorem will show that there is indeed no difference regarding the integers.

Definition 5.2.5. A functional η ∈ C′c(Z) is positive definite if

⟨η, f ∗ ˜f ⟩ ≥ 0,

for all f ∈ Cc(Z). One says a functional η ∈ C′c(Z) is (Fourier) transformable ifand only if there exists a unique measure ˆη ∈M ([0, 1)) such that

⟨η, f ∗ ˜f ⟩ = ⟨ˆη, | f ∨|2⟩for all f ∈ Cc(Z).

To see that the latter part of the definition is well-defined see [3, Ch. 2] or [13,Thm. 4.7]. Definition 5.2.5 is indeed a special case of Definition C.2.1. Equivalentto η ∈ C′c(Z) being positive definite is that the function (x ↦→ ⟨η, δx ∗ f ∗ ˜f ⟩)is positive definite for all f ∈ Cc(Z), [13, Prp. 4.4]. Note that each measureµ ∈ M (Z) induces a functional in C′c(Z) via ⟨µ, f ⟩ =

∫f dµ. We have µ ∗ ˜µ is

positive definite (in the sense of Definition 5.2.5), where ⟨˜µ, g⟩ ≔ ⟨µ, g⟩ for allg ∈ L1

µ(Z), as for all f ∈ Cc(Z)

⟨µ ∗˜µ, f ∗ ˜f ⟩ =∫ ∫f ∗ ˜f (x + y) d˜µ(x) dµ(y)

=

∫ ∫ ∫f (z)˜f (x + y − z) dωZ(z) d˜µ(x) dµ(y)

=

∫ ∫ ∫f (y + z)˜f (x − z) dωZ(z) d˜µ(x) dµ(y)


=

∫ ∫f (y + z) dµ(y)

∫ ˜f (x − z) d˜µ(x) dδZ(z).

Set gz(x) ≔ ˜f (x − z), then ˜gz(x) = ˜f (−x − z) = f (x + z). Hence

⟨µ ∗˜µ, f ∗ ˜f ⟩ =∫ ∫f (y + z) dµ(y)

∫ ˜gz(x) dµ(x) dδZ(z)

=

∫ ∫f (y + z) dµ(y)

∫f (x + z) dµ(x) dδZ(z)

=

∫ ∫f (x + z) dµ(x)

2dδZ(z) ≥ 0. (5.5)

Remark 5.2.6. By Remark C.2.4 the Bochner transform coincides with the trans-form of a functional on the integers and thus is a homeomorphism from SFP(Z) toM ([0, 1)), by Remark 5.2.4. Here M ([0, 1)) is equipped with the vague topologyand SCP(Z) with the topology of pointwise convergence, which is equivalent tothe vague topology on SFP(Z), Appendix B.2.

Before stating the theorem we note that a functional η ∈ SFP(Z) is calledtranslation bounded, if the corresponding function is bounded. For a more generaldefinition, see Definition C.1.2.

Theorem 5.2.7. The elements of SCP(Z) are bounded and the elements of SFP(Z)are translation bounded. For f ∈ C(Z), the following assertions hold

1. The Bochner transform ˆf of f , in terms of Theorem 5.2.2, exists if and onlyif f ∈ SCP(Z).

2. The Bochner transform ˆf δZ of f δZ, in terms of Definition 5.2.5, exists if andonly if f δZ ∈ SFP(Z).

3. It is f δZ ∈ SFP(Z) if and only if it exists a unique ν ∈M ([0, 1)) such that

⟨ f δZ,ˆg⟩ = ⟨ν, g⟩for all g ∈ { f ∗ ˜f : f ∈ C([0, 1))}.

Especially ˆf =ˆf δZ = ν if at least one of them exists andˆν(z) = f (−z) for all z ∈ Z.Finally note that if f = f δZ is positive definite and non-negative, then ˆf = ˆf δZ ispositive definite and non-negative.

Proof. The last assertion is by [13, Thm. 4.16]. For everything else the discussionin Appendix C.2 serves as a proof. For (3) take note of Lemma C.2.3. It shows⟨ f δZ,ˆg(−·)⟩ = ⟨ˆf δZ(−·), g⟩, which is equivalent to ⟨ f δZ, ˆg(−·)⟩ = ⟨ˆf δZ, g(−·)⟩ andhence ν =ˆf δZ. Special attention should be given to Remark C.2.4 which explainsmost of the statements made in the theorem. �


Chapter 6

Dynamical systems

Within this chapter and everything reliant, all measures are Borel probability mea-sures on (X,B) and we denote the space of all such measures by M1(X). As be-fore X denotes a locally compact Hausdorff space. A dynamical system is a triple(X,B,T ), where T : X → X is measurable. A measure theoretical dynamicalsystem is a quadruple (X,B,T, µ), where (X,B,T ) is a dynamical system andµ ∈ M1(X) is a T -invariant probability measure. Measure theoretical dynamicalsystems are called

1. ergodic if for any A ∈ B which fulfils µ(T−1(A) △ A) = 0 it holds thatµ(A) ∈ {0, 1}.

2. uniquely ergodic if there is exactly one T -invariant ergodic measure µ ∈M1(X).

3. weakly mixing if

limN→∞

1N

N−1∑n=0

µ(T−n(A) ∩ B) − µ(A)µ(B)

= 0,

for all A, B ∈ B.

4. strong mixing, if limN→∞ µ(T−N(A) ∩ B) = µ(A)µ(B), for all A, B ∈ B.

5. exact, if BN ≔⋂

n∈N T−n(B) = {∅, X} up to sets of measure zero.

They admit the following relations

exact⇒ strong mixing⇒ weakly mixing⇒ ergodic,

see e.g. [85] (after Def. 4.14 and Lemma 6.11). There are equivalent characteri-sations of ergodicity, weakly mixing and strong mixing via measurable functions,

87

88 CHAPTER 6. DYNAMICAL SYSTEMS

see [85, Thm. 1.23] or [30], which may be used occasionally throughout thiswork. For compact spaces X another structure is minimality, that is, for all x ∈ Xthe orbit of x, given by O(x) ≔ {T n(x) : n ∈ N} if T is non-invertible and in thecase T is invertible given by O(x) ≔ {T n(x) : n ∈ Z}, is dense. In this case onesays that the dynamical system (X,B,T ) is minimal or that T is minimal.

6.1 Operator theoryIn this section the necessary operator theory, which will find use in this work isintroduced.

For a locally compact Hausdorff space X, the measure space (X,B, µ), whereµ ∈ M (X), the space Lp

µ(X) = Lpµ(X,C), where 1 ≤ p ≤ ∞, of all p-times µ-

integrable functions is a vector space with norm ∥·∥p. The space Lpµ(X) is complete

and hence a Banach-space, [31, §2, Satz 2.5] or [86, p.19]. In the following we letB denote some Banach space, but the choice in most situations that will come upis B = L2

µ(X). The space

L (B) ≔ {T : B→ B : T is continuous and linear}

of continuous linear operators on A is again a Banach space with norm ∥T∥L =sup{∥T ( f )∥A : f ∈ A, ∥ f ∥B ≤ 1}, [86, Satz II.1.4]. The space L (B) is a Banachalgebra by concatenation, in terms T ◦ F, where T, F ∈ L (B) and as ∥T ◦ F∥L ≤∥T∥L ∥F∥L , [1, Satz 3.3]. The neutral element of L (B) will be denoted by Id andis referred to as the identity.

6.1.1 Spectrum of an operatorLet T ∈ L (B). The kernel of T is given by ker(T ) ≔ { f ∈ B : T ( f ) = 0}, whilewe denote the image of T by im(T ) ≔ {T ( f ) : f ∈ B}. With this the resolvent isdefined by

ResL (T ) ≔ {λ ∈ C : ker(λ Id−T ) = {0} and im(λ Id−T ) = B}

and the spectrum is denoted by

σL (T ) ≔ C\ResL (T ).

Note that for λ ∈ ResL (T ) we have that the inverse exists and (λ Id−T )−1 ∈ L (A).Also C = ResL (T ) ⊎ σL (T ), the set ResL (T ) is open, σL (T ) is compact and thevalue sup{|λ| : λ ∈ σL (T )} is called spectral radius of T , [1, Ch.9]. One has thatlimn→∞ ∥T n∥

1/nB , [86, Satz VI.1.6].

Remark 6.1.1. The formalism holds if L (B) is replaced by some Banach algebraover C with unity, which leads to Gelfand theory, see [79, Appendix] and [86,Kap. IX] for another source on that topic.

6.2. SPECTRAL MEASURES 89

6.1.2 Isometric isomorphisms or unitary operators

The case B = L2µ(X) is special, since we have a Hilbert space in this case with

scalar product given by ⟨ f , g⟩ =∫

f gµ for f , g ∈ L2µ(X). Any endomorphism

U ∈ L (L2µ(X)) which is an isometry, that is ∥U( f )∥p = ∥ f ∥p, for all f ∈ L2

µ(X)has ker(U) = {0}, and hence 0 ∈ ResL (U). Therefore U−1 ∈ L (L2

µ(X)) and fromthe parallelogram law 4⟨ f , g⟩ = ∥ f + g∥22 + ∥ f − g∥22, [86, Satz V.1.7] we deduce⟨U( f ),U(g)⟩ = ⟨ f , g⟩ and with that U−1 = U∗, the adjoint of U. Hence U is anisometric isomorphism and is also called unitary operator. The spectrum σL ofan isometric isomorphism is contained in the unit circle. For an invertible operatorwe can show that for any f ∈ L2

µ(X) the function n ↦→ ⟨Un( f ), f ⟩ on Z is positivedefinite, as for all N ∈ N it holds that

∑0≤i, j≤N

cic j⟨U i− j( f ), f ⟩ =∑

0≤i, j≤N

⟨ciU i( f ), c jU j( f )⟩ =

∑0≤i≤N

∫ciU i f dµ

2 ,

where ci ∈ C, 0 ≤ i ≤ N. This property allows us to take the Bochner transformof the function, which will be called spectral measure. This is defined in detail inthe next section.

6.2 Spectral measuresIn prospect of Section 6.1.2 the following is well-defined.

Definition 6.2.1 (Spectral measure). Let U ∈ L (L2µ(X)) be an isometric isomor-

phism. For any f ∈ L2µ(X), the unique measure ϱ = ϱ f = ϱ f (U) on [0, 1), associ-

ated with (⟨Un( f ), f ⟩)n∈Z by Bochner’s Theorem, is called the spectral measure off with respect to U.

This definition may be generalised to isometries U ∈ L (L2µ(X)). The sequence

(⟨Un( f ), f ⟩)n∈N is then only defined for non-negative n ∈ Z, but can be extendedto a positive definite sequence, by setting

un ≔

⎧⎪⎪⎨⎪⎪⎩⟨U−n( f ), f ⟩ , n < 0⟨Un( f ), f ⟩ , n ≥ 0

. (6.1)

Then, again, one can take the Bochner transform ϱ of (un)n∈Z. We will see in Sec-tion 6.5 that a way to generate a large class of isometries is given via the Koopmanoperator UT in case of measure theoretical dynamical systems (X,B,T, µ) withmeasure preserving transformations T , as ⟨ f ◦ T, g ◦ T ⟩ = ⟨ f , g⟩ for f , g ∈ L2

µ(X).


6.3 Return time combsThe results obtained here and in Section 6.5.1 appeared in [50].

There is a natural way of generating positive definite functions on Z (or equiva-lently functionals or sequences on Z) for ergodic dynamical systems, which turnsout to be similar to the sequences obtained in the construction of spectral mea-sures. At the end of this section a complete description of their relation will begiven. The different cases made throughout this section are of a technical natureand done to present a unified representation of the autocorrelation defined here.

Definition 6.3.1. Let (X,B,T, ν) be a measure theoretical dynamical system,where ν ∈ M1(X) is ergodic, let f : X → C be a ν-measurable and boundedfunction (especially f ∈ L2

µ(X)) and let y ∈ X. The discrete comb on Z

ηy ≔

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩∑n∈N

f ◦ T n(y) δn, if T is non-invertible,∑z∈Z

f ◦ T z(y) δz, if T is invertible.

is called f -weighted return time comb ηy with respect to T and reference point y.

For practical reasons the definition will be often shortened to return time comb.In case the following vague limits of a weighted return time comb ηy

v-limn→∞

ηy |[−n,n] ∗ ˜ηy |[−n,n]

n + 1, if T is non-invertible,

v-limn→∞

ηy |[−n,n] ∗ ˜ηy |[−n,n]

2n + 1, if T is invertible.

exist, it is called the autocorrelation of ηy and is denoted by γy or ηy ~ ˜ηy to em-phasise the involvement of a convolution. Note that in the non-invertible caseηy |[−n,n] = ηy |[0,n]. Further, for each n ∈ N the measure ηy |[−n,n] ∗ ˜ηy |[−n,n] is positivedefinite by (5.4) in Section 5.2. As the space of positive definite measures forms aclosed convex cone, [13, Prp. 3.6(i)], we conclude that γy is always a positive def-inite measure. To have a more convenient way of stating the upcoming theorem,we define a positive definite sequence (Ξ(T, ν)(z))z∈Z on Z, by

Ξ(T, ν)(z) ≔

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫f ◦ T z · f dν = ⟨ f ◦ T |z|, f ⟩, if T is non-invertible, z ≥ 0

∫f ◦ T−z · f dν = ⟨ f ◦ T |z|, f ⟩, if T is non-invertible, z < 0

∫f ◦ T−z · f dν = ⟨ f ◦ T z, f ⟩, if T is invertible

6.3. RETURN TIME COMBS 91

If f is real valued the representation of Ξ shortens to

Ξ(T, ν)(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∫f ◦ T |z| · f dν = ⟨ f ◦ T |z|, f ⟩ if T is non-invertible,

∫f ◦ T−z · f dν = ⟨ f ◦ T z, f ⟩ if T is invertible.

Theorem 6.3.2. For an f -weighted return time comb ηy with respect to T and withreference point y, the autocorrelation γy exists for ν-almost every y and equals

γηy =∑z∈Z

Ξ(T, ν)(z) δz.

Proof. We will prove the statement in the case that T is non-invertible. The casewhen T is invertible follows analogously. For every ϕ ∈ Cc(Z), the result in [80,Lem. 1.2] yields

limN→∞

1N + 1

(ηy |[−N,N] ∗ ˜ηy |[−N,N] − ηy ∗ ˜ηy |[−N,N]

)= 0,

for which it is needed that ηy is a translation bounded measure. Hence, for everyϕ ∈ Cc(Z),

⟨γy, ϕ⟩ = limN→∞

1N + 1

⟨ηy ∗ ˜ηy |[−N,N], ϕ

⟩= lim

N→∞

1N + 1

∫ ∫1[−N,N](n) ϕ(m + n) d˜ηy(n) dηy(m)

= limN→∞

1N + 1

∫ ∫1[−N,N](−n) ϕ(m − n) dηy(n) dηy(m)

= limN→∞

1N + 1

∑m∈N0

N∑n=0

ϕ(m − n) f ◦ T n(y) · f ◦ T m(y)

= limN→∞

1N + 1

∑z∈Z

ϕ(z)∑−z≤n≤N

f ◦ T n(y) · f ◦ T n+z(y).

Noticing that if z < 0, then

f ◦ T n · f ◦ T n+z =(

f ◦ T−z · f)◦ T n+z

and if z ≥ 0, then

f ◦ T n · f ◦ T n+z =(

f · f ◦ T z)◦ T n


allows us to split up the sum as follows

γy = limN→∞

1N + 1

⎛⎜⎜⎜⎜⎜⎜⎝ ∑z<0

ϕ(z)∑−z≤n≤N

(f ◦ T−z · f

)(T n+z(y))

+∑z≥0

ϕ(z)∑

0≤n≤N

(f · f ◦ T z

)(T n(y))

⎞⎟⎟⎟⎟⎟⎟⎠=

⎛⎜⎜⎜⎜⎜⎜⎝ ∑z<0

ϕ(z) limN→∞

N + z + 1N + 1

1N + z + 1

∑−z≤n≤N

(f ◦ T−z · f

)(T n+z(y))

+∑z≥0

ϕ(z) limN→∞

1N + 1

∑0≤n≤N

(f · f ◦ T z

)(T n(y))

⎞⎟⎟⎟⎟⎟⎟⎠ . (6.2)

An application of Birkhoffs ergodic theorem yields∑z∈Z

Ξ(T, ν)(z) δz =∑z<0

ϕ(z)∫

f ◦ T−z · f dν +∑z≥0

ϕ(z)∫

f · f ◦ T z dν.

�

As γy is always a positive definite measure the upcoming is well-defined.

Definition 6.3.3 (Spectral return comb). For an f -weighted return time comb ηy

with respect to T and reference point y, the Bochner transform ˆγy ∈M ([0, 1)) ofγy with respect to Theorem 5.2.7 is called f -weighted spectral return time measureηy with respect to T and reference point y or short spectral return measure.

We want to remark that the spectral return measure ˆγy has its total mass givenby ˆγy([0, 1)) = γy(0) =

∫| f |2 dν. As the Bochner transform is a continuous

operator, see [79, Ch. D, D.5, Thm.] or [13, Thm. 3.13], the limits in the definitionof the autocorrelation can be taken after taking the transform. For that let γy bethe autocorrelation of a weighted return time comb ηy for an ergodic dynamicalsystem (X,B,T, ν). Let c = 2 if T is invertible and c = 1 otherwise, then itν-almost surely holds that

ˆγy =

⎛⎜⎜⎜⎜⎝v-limn→∞

ηy |[−n,n] ∗ ˜ηy |[−n,n]

cn + 1

⎞⎟⎟⎟⎟⎠⋀= v-lim

n→∞

1cn + 1

(ηy |[−n,n] ∗ ˜ηy |[−n,n]

)∧=v-lim

n→∞

1cn + 1

(ηy |[−n,n]

)∧∗(˜ηy |[−n,n]

)∧= v-lim

n→∞

1cn + 1

(ηy |[−n,n]

)∧ 2. (6.3)

In the following the relation to spectral measures in terms of (6.1) implied by Ξwill be established, which builds up a nice relation between spectral measures and


spectral return measures. In fact they are equal if we average with respect to ν,more precisely it holds∫

Xγy(z) dν(y) =

⎧⎪⎪⎨⎪⎪⎩⟨U−z( f ), f ⟩ , z < 0⟨Uz( f ), f ⟩ , z ≥ 0

, (6.4)

where z ∈ Z. An autocorrelation for return time combs could have been definedin this way, but would trade one technicality for another, namely reference pointswith respect to ν for integration with respect to ν. In case of unique ergodicityand continuous bounded functions f they are equal for all reference points y ∈ X.We will always consider our quasicrystals on the integers, while in Chapter 7 atreatment for quasicrystals on more general spaces is discussed. The integers arewell suited to give us interesting insights presented in Section 6.5.1 and 8.4. Otherworks which emphasise the relation between diffraction and spectral measures are[8, 58]. A special case are subshifts emerging from primitive substitutions, whichare introduced in Section 6.6. Due to unique ergodicity, the approaches fromquasicrystals, return time combs and spectral measures coincide in this case. Thediffraction is then also called correlation measure and a nice description is givenin [71, Ch. 4.3.1]. The last part of this section deals with the case when differentweight functions are considered.

Definition 6.3.4. Let η1, η2 ∈ C′c(Z). If the sequence 1

N+1 ((η1)|[−N,N] ∗ (η2)|[−N,N])given for N ∈ N attains a vague limit one writes

η1 ~ η2 ≔ v-limN→∞

1N + 1

((η1)|[−N,N] ∗ (η2)|[−N,N]

)and says that the averaged convolution of η1 and η2 exists.

Corollary 6.3.5. If ηy is a f -weighted return time comb and η′y a g-weighted returntime comb, both with respect to an ergodic system (X,B,T, ν), the measure ηy~˜η′yexists for ν-almost every y ∈ X and is given by

ηy ~ ˜η′y(z) =

⎧⎪⎪⎨⎪⎪⎩∫

f ◦ T z · g dν, z ≥ 0∫f · g ◦ T−z dν, z < 0

.

Additionally the Bochner transform of ηy ~ ˜η′y is for ν-almost every y ∈ X well-defined.

Proof. The first part of the corollary is literally the same proof as the one of Theo-rem 6.3.2, where one f is replaced by g. To see that the Bochner transform exists,first confirm that ηy |[−N,N] ∗

˜η′y |[−N,N]can be seen as a function of SCP(Z) by The-

orem 5.2.7 and Equation (5.3) for all N ∈ N. Hence ηy |[−N,N] ∗˜η′y |[−N,N]

is a linearcombination of positive definite functions. As the space of positive definite func-tions is closed, [13, Prp. 3.6(i)], the Bochner transform is well-defined wheneverthe limit exists. �


6.3.1 Rotations on the unit circleOne of the simplest examples of dynamical systems as well as lattices, periodicstructures, or aperiodicity is given by periodic, respectively aperiodic, rotationson the unit circle. This section is the first in this work that introduces examplesfor which the Bochner transform will be calculated explicitly. As shown in Sec-tion 7.1.1, this also fits into the setting of [7, 73] and we will see that indeed theresults shown here are the same as one would expect for Cut and Project Schemesand for spectral measures.

For α > 0, we define Tα : [0, 1) → [0, 1) by Tα(x) ≔ {x + α} and con-sider the dynamical system ([0, 1),B,Tα). In the case that α is irrational, thetransformation Tα is uniquely ergodic, where the unique ergodic measure is theLebesgue measure m|[0,1) ≕ µα. On the other hand, if α = p/q with p, q ∈ N andgcd(p, q) = 1, for each w ∈ [0, 1), the measure µα ≔ µq,w ≔ q−1 ∑

k∈Zqδ{w+k/q} is

an ergodic measure for Tα. In both cases Tα is not mixing with respect to µα, asby [85, Thm. 1.27] no translation map on a compact group is weakly mixing.

Let α = p/q with p, q ∈ N and gcd(p, q) = 1. For w ∈ [0, 1) we havesupp(µq,w) ≅ Zq and Ξ(T, µq,w)(z) = Ξ(Tα, µq,w)([z]q) for z ∈ Z. Further, for everyy ∈ supp(µq,w), Theorem 6.3.2 yields

γy =∑z∈Z

Ξ(Tα, µq,w)(z) δz, (6.5)

where ηy denotes the f -weighted return time comb with respect to Tα and withreference point y. If α ∈ R≥0 \Q, then supp(µα) = supp(m|[0,1)) = [0, 1), combiningthis with Theorem 6.3.2 yields, for m|[0,1)-almost every y,

γy =∑z∈Z

Ξ(Tα,m|[0,1))(z) δz, (6.6)

where again ηy denotes the f -weighted return time comb with respect to Tα andwith reference point y.

Remark 6.3.6. If α is irrational and f is Riemann integrable, then the autocor-relation exists for every reference point y and is independent of y. This followsby using an approximation argument due to equi-distribution of the orbit of yand the fact that Tα is uniquely ergodic, see [54, 85, Exl. 2.1,Thm. 6.18]. Dueto the structure of Tα, this result is in line with those of [75], where one takes(R, [0, 1), {(n, {nα}) : n ∈ Z}) as the cut and project scheme, see Section 7.1.1.Further, as we will shortly see in (6.9),(6.10), the weights Ξ(Tα, ·) are given by aconvolution of certain functions. Note that Riemann integrability of a function fon [0, 1) is understood Riemann integrability of any arbitrary extension of f onto[0, 1], see [78, Ch. 6] or [76, Thm. 11.1.6]. In particular this implies that anyRiemann integrable function f on [0, 1) is bounded.


Theorem 6.3.7. Let α > 0 and w ∈ [0, 1). For a given y ∈ [0, 1), let ηy denote thef -weighted return time comb with respect to Tα and with reference point y.

1. If α = p/q with p, q ∈ N and gcd(p, q) = 1, then for every reference pointy ∈ supp(µq,w), the spectral return measure of ηy is given by

ˆγy =∑m∈Zq

ˆΞ(Tα, µq,w)(m) δ{mα} =∑m∈Zq

|ˆfα,y|2(m) δ{mα},

where fα,y(k) ≔ f (T kα(y)) for k ∈ Zq.

2. If α ∈ R≥0 \ Q, then the spectral return measure of ηy is, for m|[0,1)-almostevery reference points y ∈ [0, 1), given by

ˆγy =∑m∈Z

ˆΞ(Tα,m|[0,1))(m) δ{mα} =∑m∈Z

|ˆf |2(m) δ{mα}.

Additionally, if f is Riemann integrable, then the statement holds for everyreference point y ∈ [0, 1).

Proof. First we consider rational values for α. By definition it holds that

Ξ(Tα, µq,w)(k) =∫

f ◦ T−kα · f dµq,w

=

∫fα,y(l − k) · fα,y(l) dωZq(l) = fα,y ∗ ˜fα,y(k),

(6.7)

and so ˆΞ(Tα, µq,w)(m) = ( fα,y ∗ ˜fα,y)∧(m) = |ˆfα,y|2(m). To have a better distinctionbetween rational and irrational α let (·, ·)α : Zq×Zq → [0, 1) be defined by (k, z)α ≔exp(2πi α k z), and let (·, ·) : [0, 1) × Z → S1 be defined by (x, z) ≔ exp(2πixz).Letting ϕ ≔ ϕ1 ∗ ˜ϕ1, where ϕ1 ∈ Cc([0, 1)), we observe the following chain ofequalities.∑

z∈Z

Ξ(Tα, µq,w)(z) ˆϕ(z) =∑z∈Z

∑k∈Zq

|ˆfα,y|2(k) (k, z)α ˆϕ(z)

=∑k∈Zq

|ˆfα,y|2(k)∑z∈Z

ˆϕ(z) ({kα}, z) =∑k∈Zq

|ˆfα,y|2(k) ϕ(kα)

In the third equality we have used Lebesgue’s dominated convergence theorem.For this, observe that the function |ˆfα,y|2 is bounded and by the fourier representa-tion for functions limN→∞

∑Nz=−N ˆϕ(z) (k{α}, z) = ϕ(kα), which is bounded for all

k ∈ Z, since ϕ is a continuous function on [0, 1). As a result

|ˆfα,y|2(k)

∑−N≤z≤N

ˆϕ(z) (αk, z)

≤|ˆfα,y|2(k)

∑−N≤z≤N

|ˆϕ|(z)


=|ˆfα,y|2(k)∑−N≤z≤N

ˆϕ(z)

=|ˆfα,y|2(k)∑−N≤z≤N

ˆϕ(z)(1, z) ≤ |ˆfα,y|2(k)ϕ(0),

has a global bound, for all k and N. Hence ⟨γy,ˆϕ⟩ = ⟨ˆγy, ϕ⟩, which completes theproof of Part 1.

Part 2 follows analogously to Part 1, where one replaces µα,y by µα, (k, z)α by({kα}, z) and fα,y by f . In particular,

Ξ(Tα, µα)(k) =∫

f ◦ T−kα · f dµα

=

∫f (x + kα) · f (x) dm|[0,1)(x) = f ∗ ˜f (kα).

(6.8)

The final statement follows by using identical arguments to those given in Re-mark 6.3.6. �

Here we emphasise that, for w ∈ [0, 1) and α > 0 irrational, when f ∗ ˜f (zα)is written, it denotes the convolution with respect to µα = m|[0,1) evaluated at{zα} ∈ [0, 1), for z ∈ Z. In the case that α = p/q with p, q ∈ N and gcd(p, q) = 1the convolution with respect to µq,w evaluated at k ∈ Zq, and w ∈ [0, 1) is denotedby fα,w ∗˜fα,w(k). Namely,

f ∗ ˜f (kα) =∫

f ◦ T−kα · f dm|[0,1) = Ξ(Tα,m|[0,1))(k), (6.9)

and

fα,w ∗˜fα,w(k) =∫

[0,1)f ◦ T−k

α · f dµα,w = Ξ(Tα, µq,w)(k). (6.10)

In what follows, for α ∈ R+ and a sequence (αi)i∈N in R+ which converges to α, werequire that the sequence (αi)i∈N does not attain the value α infinitely often. Thiscondition is important as in the following lemma, if α = p/q with p, q ∈ N andgcd(p, q) = 1, we show that the pointwise limit of Ξ(Tαi , µαi) does not coincidewith Ξ(Tα, µq,w), for any w ∈ [0, 1).

The remainder of this section considers non-negative functions f for weightedreturn time combs, instead of complex-valued functions to ease the properties forRiemann integration. Be noted that in the case of complex-valued functions f ,one can consider Riemann integration for Banach spaces as presented in [2].

Lemma 6.3.8. Let f : [0, 1) → R≥0 be Riemann integrable, y ∈ [0, 1), α ∈ R+

and (αi)i∈N a sequence in R+ that converges to α with αi ≠ α for all i ∈ N. Thesequence of functions given by Ξ(Tαi , µαi,y) if αi is rational, and by Ξ(Tαi ,m|[0,1)),if αi is irrational, converge uniformly to Ξ(Tα,m|[0,1)).


Proof. In the case that αi ∉ Q for all i ∈ N, the result is a consequence of the factthat f ∗ ˜f : [0, 1)→ R is a continuous function on the compact group ([0, 1),+) ≅([0, 1),+). On the other hand, if αi = pi/qi with pi, qi ∈ N and gcd(pi, qi) = 1, forall i ∈ N, then observe that

f ∗ ˜f (zα)− fαi,y ∗˜fαi,y([z]qi)

≤

f ∗ ˜f (zα) − f ∗ ˜f (zα)

+

f ∗ ˜f (zαi) − fαi,y ∗

˜fαi,y([z]qi). (6.11)

The first term on the right hand side of (6.11) converges to zero, by an analogousargument to that given when αi ∉ Q for all i ∈ N. To complete the proof, we showthat the second term on the right-hand-side of (6.11) also converges to zero. Here,we will make use of the Riemann integrability of f . Let i ∈ N, yi = min supp(µq,y)and z ∈ Z. Setting l = [αiz]qi and

Im ≔

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩[yi +

mqi, yi +

m+1qi

]if m ∈ {0, 1, . . . , qi − 2},

[0, yi) ∪[yi +

qi−1qi, 1

]if m = qi − 1,

we havef ∗ ˜f (zαi) − fαi,yi ∗

˜fαi,yi([z]qi)

=

∫ f (x) · f

({x − l

qi

})dm|[0,1)(x) − q−1

i

∑m∈Zqi

f({

yi +mqi

})· f

({yi +

m−lqi

})=

∑m∈Zqi

∫1Im

(x)(

f (x) · f({

x − lqi

})− f

({yi +

mqi

})· f

({yi +

mqi− l

qi

}))dm|[0,1)(x)

=

∑m∈Zqi

∫1Im

(x)(

f (x)(

f({

x − lqi

})− f

({yi +

mqi− l

qi

}))+ f

({yi +

mqi− l

qi

}) (f (x) − f

({yi +

mqi

})))dm|[0,1)(x)

=

∑m∈Zqi

∫1I[m−l]qi

(x) · f({

x + lqi

}) (f (x) − f

({yi +

m−lqi

}))dm|[0,1)(x)

+

∫1Im

(x) · f({

yi +m−lqi

}) (f (x) − f

({yi +

mqi

}))dm|[0,1)(x)

≤ ∥ f ∥∞

∑m∈Zqi

q−1i

(sup

{f (x) : x ∈ I[m−l]qi

}− inf

{f (x) : x ∈ I[m−l]qi

}+ sup { f (x) : x ∈ Im} − inf { f (x) : x ∈ Im}

)


= 2 ∥ f ∥∞∑

m∈Zqi

q−1i

(sup { f (x) : x ∈ Im} − inf { f (x) : x ∈ Im}

).

Since limi→∞ qi = ∞, this latter term converges to zero by the Riemann prop-erty of Darboux. Moreover, this latter term is independent of z, yielding uniformconvergence. �

Lemma 6.3.9. Let f : [0, 1) → R≥0 be Riemann integrable, α ∈ [0, 1) and (αi)i∈N

a sequence in R+ such that limi→∞ αi = α with αi ≠ α, for all i ∈ N. Let (yi)i∈N

denote a sequence of reference points in [0, 1) and, for i ∈ N, let µyi denote thef -weighted return time comb with respect to Tαi and with reference point yi. Thesequence of autocorrelations (γµyi

)i∈N attains a vague limit γ given by

γ =∑z∈Z

Ξ(Tα,m|[0,1)) δz.

Proof. If αi is irrational the result is a direct consequence of (6.6), (6.8) andLemma 6.3.8, since f ∗ ˜f is independent of the starting point yi. If αi = pi/qi withpi, qi ∈ N and gcd(pi, qi) = 1 for all i ∈ N, then the result follows from (6.5), (6.7)and an analogous argument for the one given in the proof of Lemma 6.3.8. �

The fact that the Bochner transform of a sequence of autocorrelations con-verges, if the sequence of autocorrelations itself converges is illustrated in Fig-ure 6.1, which shows the spectral return measures of two f -weighted return timecombs associated with irrational rotation numbers. The explicit values of thevague limit will be determined in the following.

Figure 6.1: The 50 largest atoms ofˆγ for the (1/√

2) Id-weighted return time combwith number α = π/20 (dots) and α = 103π/2000 (triangles).


Theorem 6.3.10. Let f : [0, 1)→ R≥0 be Riemann integrable and let α ∈ R+. Fixa sequence ( fi)i∈N of non-negative Riemann integrable functions that convergesuniformly to f on [0, 1), and fix a sequence (αi)i∈N in R+ with limi→∞ αi = α andαi ≠ α for all i ∈ N. Let (yi)i∈N denote a sequence of reference points in [0, 1) and,for i ∈ N, let µyi denote the fi-weighted return time comb with respect to Tαi andwith reference point yi. The sequence of autocorrelations (γµyi

)i∈N attains a vaguelimit γ given by

γ =∑z∈Z

Ξ(Tα,m|[0,1))(z) δz

and

ˆγ =∑m∈Z

ˆΞ(Tα,m|[0,1))(m) δ{mα}.

In particular ˆΞ(Tα,m|[0,1))(m) = |ˆf |2(m) for all m ∈ N.

Proof of Theorem 6.3.10. If αi ∉ Q for all i ∈ N, then the convergence followsfrom (6.6), (6.8) and the fact that ( fi) ∗ ˜( fi) converges to f ∗ ˜f uniformly – this factis a direct consequence of how the involved maps are defined.

If αi = pi/qi with pi, qi ∈ N and gcd(pi, qi) = 1 for all i ∈ N, then as fi

converges to f uniformly, given ε > 0 there exists N ∈ N such that ∥ fi − f ∥∞ < εfor all i ≥ N. Hence it holds for all i ≥ N that

f ∗ ˜f − ( fi)αi,yi ∗˜( fi)αi,yi

=

f ∗ ˜f − fαi,yi ∗

˜fαi,yi + fαi,yi ∗˜fαi,yi

− ( fi)αi,yi ∗˜fαi,yi + ( fi)αi,yi ∗

˜fαi,yi − ( fi)αi,yi ∗˜( fi)αi,yi

≤


˜fαi,yi

+

( fαi,yi − ( fi)αi,yi) ∗ ˜fαi,yi

+

( fi)αi,yi ∗ ( ˜fαi,yi −

˜( fi)αi,yi)

≤


˜fαi,yi

+

2qi

∑m∈Zqi

(∥ f ∥∞ + ε) ∥ fi − f ∥∞

≤


˜fαi,yi

+ 2 (∥ f ∥∞ + ε) ε.

This together with (6.5), (6.7) and Lemma 6.3.9 yields the result. That ˆγ =∑m∈Z

ˆΞ(Tα,m|[0,1))(m) δ{mα} is by Theorem 6.3.7(2). Even if α ∈ Q this still holdsdue to |(( fi)αi,yi)

∧|2 → |ˆf |2 for i → ∞ by continuity of the fourier transform as amap from L1

m to C0. �

Remark 6.3.11. Note that the case α = 0 is nothing special, as for the sys-tem ([0, 1),B,T0), δy), where T0 = id, y ∈ [0, 1), the map Ξ(T0,m|[0,1))(z) =

100 CHAPTER 6. DYNAMICAL SYSTEMS∫f 2 dm|[0,1) for all z ∈ Z. Thus for the autocorrelation γ with respect to Ξ the spec-

tral return measure ˆγ = ∫f 2 dm|[0,1) δ0, which is consistent with Theorem 6.3.10

in the case α = 0 and hence we may assume α ∈ R≥0.

Remark 6.3.12. In prospect to the results of Beckus and Pogorzelski in [10], theconditions of Theorem 6.3.10 are met, if the sequence (αi)i∈N is a sequence ofirrational numbers and α is also an irrational number. Here, the authors developedweak-∗-convergence results for autocorrelations of approximations of uniquelyergodic Delone dynamical systems. That such behaviour cannot be expected fornon-uniquely ergodic Delone dynamical systems can be seen for rational choicesof α as follows. Let f : [0, 1) → R≥0 be continuous and bounded, fix a referencepoint y ∈ [0, 1), let α > 0 be rational and let (αi)i∈N denote a fixed sequence in R+converging to α. In this case, it is evident that the f -weighted return time combswith respect to Tαi and reference point y converges to the f -weighted return timecomb with respect to Tα and reference point y. However, the associated sequenceof autocorrelations may not converge. Specific examples demonstrating this aregiven below.

Example 6.3.13. Let α = 1/q with q ≥ 3 an integer and let y = 0. If f (x) =2qx(1 − qx/2)1[0,2/q), then for any sequence (αi)i∈N in R+ of irrational numberswith limi→∞ αi = α and z ∈ Z, we have

limi→∞Ξ(Tαi ,m|[0,1))(z) = f ∗ ˜f (zα)

=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩(q4{αz}5 − 10q3{αz}4 + 20q2y3)/30 if 0 < {αz} < 2/q,−(q{αz} − 4)3(q2{αz}2 + 2q{αz} − 4)/(30q) if 2/q ≤ {αz} < 4/q,0 otherwise.

=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩11/(30q) if [z]q ∈ {1, 3},16/(15q) if [z]q = 2,0 otherwise.

On the other hand,

Ξ(Tα, µq,0)(z) = q−1∑k∈Zq

f(

kq

)· f

({ k−[z]q

q

})=

⎧⎪⎪⎨⎪⎪⎩1/q if [z]q = 2,0 otherwise.

By the continuity of f , we have additionally that limi→∞ µαi,y = µα,y. This showsthat, in general, the autocorrelation is not continuous as a function of the returntime comb.

6.4. SPECTRUM FOR ISOMETRIC ISOMORPHISMS 101

Example 6.3.14. Let α = p/q with p, q ∈ N and gcd(p, q) = 1. If f = 1[0, rq ) for

a fixed r ∈ Zq, then for any sequence (αi)i∈N ∈ [0, 1)N with limi→∞ αi = α impliesthat, for all z ∈ Z,

limi→∞Ξ(Tαi ,m|[0,1))(z) = Ξ(Tα, µq,0)(z).

This is due to the fact that, for m ∈ Zq,∫q · f · 1[ m−1

q , mq ) dm|[0,1) = f (m−1

q ).

Hence, for all m ∈ Zq,

Ξ(Tα,m|[0,1))(m) = Ξ(Tα, µq,0)(m).

Since f ∗ ˜f is a continuous function, if limi→∞ αi = α, then, for all z ∈ Z,

limi→∞Ξ(Tαi ,m|[0,1))(z) = f ∗ ˜f (zα) = f ∗ ˜f ({zα}) = fα,0 ∗˜fα,0([z]q) = Ξ(Tα, µq,0)(z).

Additionally, if for all i sufficiently large αi ≥ α, then limi→∞ µαi,y = µα,y.

6.4 Spectrum for isometric isomorphismsAn important feature of the spectral theorem, which will be given in the following,is given by the connection between the decomposition of isometric isomorphismsand spectral measures that will be presented here.

Theorem 6.4.1 ([71, Thm: 2.3]). Let H be a separable Hilbert space and U : H →H be an isometric isomorphism. Then, there exists a sequence (hi)i∈I ∈ HI , whereI is countable and may even be finite with the following: If we define subspacesof H by Hi ≔ span{Un(hi) : n ∈ Z}, then H =

⨁i∈I Hi and Hi ⊥ H j for i ≠ j.

Moreover for the associated sequence (ϱhi)i∈I of spectral measures we have ϱh1 ≫

ϱh2 ≫ ϱh3 ≫ . . . (after a possible re-enumeration of I). The choice of (hi)i∈I isunique up to equivalence of the spectral measures. That is, for another sequence(h′i)i∈I with the properties above, we have ϱhi ∼ ϱh′i for all i ∈ I.

One important part of the theorem is that any other measure of the sequence isabsolutely continuous to ϱh1 . This unique property will be further studied for therest of this section.

Definition 6.4.2 (Maximal spectral type). The equivalence between measures in-duces an equivalence relation on M ([0, 1)). For every isometric isomorphism Uon H, its maximal spectral type is given by the equivalence class [ϱh1], where h1

is from Theorem 6.4.1 after re-enumeration. We set ϱmax(U) = ϱmax ≔ ϱh1 .


Definition 6.4.3. The isometric isomorphism U has discrete, singular, and Le-besgue spectrum, if ϱmax is discrete, singular to the Lebesgue measure and equiv-alent to the Lebesgue measure, respectively. See 4.4.4 for the definition of theproperties for measures. Further U has simple spectrum, if ϱhi = 0 for all i ≥ 2,hence ϱmax = ϱh1 contains all the information. If ϱhi ≠ 0 for all i ∈ N+, we say thatU has countable spectrum.

Proposition 6.4.4. Let U be an isometric isomorphism on H and ϱmax be themaximal spectral type of U. Then

1. ϱ f ≪ ϱmax for all f ∈ H.

2. ϱmax = ϱh for some h ∈ H.

3. For any ϱ ∈ M+([0, 1)) with ϱ ≪ ϱmax, there exists an f ∈ H such thatϱ = ϱ f .

4. eiλ is an Eigenvalue of U if and only if ϱmax({λ}) ≠ 0.

5. σL (U) = supp(ϱmax).

Proof. [71, Cor. 2.7, 2.8, Prp. 2.14, Rem. 2.4] �

With that, a link between the spectrum σL (U) and the spectral measures ϱ f ,f ∈ H of U is given by the following relation

supp(ϱ f ) ⊆ supp(ϱmax) = σL (U).

6.5 Operator on dynamical systemsIn Section 6.1.2 the spectral theory for operators was developed and a decom-position in terms of spectral measures for isometric isomorphisms was given inTheorem 6.4.1. In this section epimorphisms are considered instead, which willsoon result in a rather restricted setting. For the Perron Frobenius operator thatwill be defined here we will always assume a potential function given by the slopeof the dynamics of the system, whether in the source [40, 51] different potentialsare considered. Later on it will be assumed that X = [0, 1], which implies forBorel measures µ, L∞µ ([0, 1]) ⊆ L2

µ([0, 1]) ⊆ L1µ([0, 1]) and the space of functions

of bounded variation BV ⊆ L∞m ([0, 1]) will be considered. The given definition

var( f ) ≔ infg= f a.s.

sup

⎧⎪⎪⎨⎪⎪⎩ ∑1≤i≤N

|g(xi) − g(xi−1)| : N ∈ N+ and 0 ≤ x0 < x1 < . . . < xN ≤ 1

⎫⎪⎪⎬⎪⎪⎭for a function f ∈ L1

m([0, 1]) is called variation of f .

6.5. OPERATOR ON DYNAMICAL SYSTEMS 103

Proposition 6.5.1. The space

BV ≔{f ∈ L1

m([0, 1],R) : var( f ) < ∞}

of functions of bounded variation, together with ∥ f ∥BV ≔ ∥ f ∥1 + var( f ), wheref ∈ BV, is a normed Banach-space. If f , g ∈ BV, then f · g ∈ BV and BV ⊆ L∞m ,(BV, ∥ · ∥BV) is dense in (L1

m([0, 1]), ∥ · ∥1) and { f ∈ BV : ∥ f ∥BV ≤ 1} is a compactsubset of (L1

m([0, 1]), ∥ · ∥1).

Proof. Almost all claims are stated in [40, Lem. 5] except BV ⊆ L∞m and f ·g ∈ BVfor f , g ∈ BV. The first follows by contra-position. For any f ∉ L∞m one hasvar( f ) = ∞, hence f ∉ BV. For the latter one observe

| f (y)g(y) − f (x)g(x)| = | f (y)g(y) − f (y)g(x) + f (y)g(x) − f (x)g(x)|≤ | f (y)| · |g(y) − g(x)| + |g(x)| · | f (y) − f (x)|≤ ∥ f ∥∞ · |g(y) − g(x)| + ∥g∥∞ · | f (y) − f (x)|,

for m-almost every x, y and hence ∥ f · g∥BV ≤ ∥ f ∥∞ · ∥g∥BV + ∥g∥∞ · ∥ f ∥BV. �

The following is based on based on [51]. We will encounter the transfer oper-ator and derive from it the Koopman, Ruelle and Perron-Frobenius operator.

Let (X,B, µ) be a measure space and T : X → X a map with µ ◦ T−1 ≪ µ.Such maps are called non-singular transformations and by the Radon-Nikodymtheorem 4.4.2, the following operator

Tµ : L1µ(X)→ L1

µ(X), Tµ(g) ≔∂(µg ◦ T−1

)∂µ

=∂µg ◦ T−1

∂µ,

where µg(A) ≔∫

Ag dµ for g ∈ L1

µ(X), is well-defined. The operator Tµ is generallyreferred to as the transfer operator of T with respect to µ. For any g ∈ L1

µ(X),from

∫1A · Tµ(g) dµ =

∫1A ◦ T · g dµ we conclude that f ∈ L∞µ (X) may be

approximated by simple functions (see e.g. [31, Def. 4.12], [77, Ch. 1]) andhence

∫f · Tµ(g) dµ =

∫f ◦ T · g dµ as discussed in [51, (2.4)]. We have

just calculated what is known as the Koopman operator UT : L∞µ (X) → L∞µ (X),f ↦→ f ◦ T associated with Tµ : L1

µ(X)→ L1µ(X) by the equality∫

f ◦ T · g dµ =∫

f · Tµ(g) dµ

where f ∈ L∞µ (X) and g ∈ L1µ(X).

Definition 6.5.2 (Dynamical spectrum). Let (X,B,T, µ) be a measure theoreticaldynamical system. Then UT : L2

µ(X)→ L2µ(X) is a continuous linear operator. The

spectrum σL (UT ) is called dynamical spectrum of T or dynamical spectrum of(X,B,T, µ). In the case that all eigenfunctions of UT are constant one says that Thas continuous spectrum.


Lemma 6.5.3. The measure theoretical dynamical system (X,B,T, µ) is weaklymixing, if and only if UT has exactly one eigenvalue, which is given by 1 and alleigenfunctions are constant.

Proof. [30, Thm. 2.36(5)] and [85, Thm. 1.26] for a detailed explanation forλ = 1. �

In the case, where X = [0, 1] and µ is the Lebesgue measure, namely µ =m, the transfer operator is close to the common understanding of integration bysubstitution, see e.g. [53, p. 204], [87, Satz IV.8.9], which will be specified in thefollowing. We do not consider T to be invertible in general, but we will considerpiecewise strictly monotone differentiable transformations T with 1 < inf |T ′| < Cfor some C > 0. In the following, maps for a partition of the domain of T will bedefined and it will be dealt with the ambiguous question concerning what happensat the endpoints of each part in terms of the derivative these maps.

Definition 6.5.4. Let X = [0, 1]. A transformation T : [0, 1] → [0, 1] admitsinverse branches, if there is a finite partition of [0, 1] into intervals (Xi)i∈I , thatis Xi ∩ X j = ∅ for i, j ∈ I with i ≠ j and [0, 1] =

⋃i∈I Xi, on which T |X◦i is

strictly monotone and differentiable for all i ∈ I. Furthermore it is assumed that∂−1T ∈ BV, which is given in the following.Define for all i ∈ I the maps Ti : Xi → T (Xi) by Ti ≔ T |Xi , they are invert-ible and T−1

i are called the inverse branches of T . For X0 ≔ [0, 1]\⋃

X◦i weset ∂−1T (x) ≔ 1/|T ′(x)| = |(T−1

i )′(T (x))| for all x ∈ [0, 1]\X0 ∩ Xi, i ∈ I and∂−1T (x) ≔ lim infy→x,y∈X\X0 1/|T ′(y)| for all x ∈ X0. In this way the derivative ofT−1

i can be extended onto the whole set Xi.

Remark 6.5.5. The strict monotonicity assumption on T , together with ∂−1T ∈BV, implies that there are constants d0, d1 ∈ R that satisfy the following chain ofinequalities 1 < d0 ≤ infx∈X\X0 |T

′(x)| ≤ supx∈X\X0|T ′(x)| ≤ d1. Especially one has

∂−1T : [0, 1]→ [1/d1, 1/d0].

If T admits inverse branches the following holds for all g ∈ L1m([0, 1])∫

ATm(g) dm =

∫1A ◦ T · g dm

=∑i∈I

∫T−1

i (T (Xi))1A ◦ T · g dm

=∑i∈I

∫T (Xi)

(1A ◦ T ◦ T−1i ) · (g ◦ T−1

i · |(T−1i )′|) dm

=

∫A

∑i∈I

g ◦ T−1i · |(T

−1i )′| · 1T (Xi) dm,


where A ∈ B. The almost sure derived equality for Tm will be referred to asPerron-Frobenius operator of T , see [70, Ch. 11.2] for more information denotedby P : L1

m([0, 1])→ L1m([0, 1]),

P(g)(x) ≔∑i∈I

g ◦ T−1i (x) · ∂−1T (T−1

i (x)) · 1T (Xi)(x)

=∑

y∈T−1(x)

g(y) · ∂−1T (y).(6.12)

Remark 6.5.6. A big contribution on the assumption on T is made in [88]. There,the author generalised the results from [56] to the class of piecewise strictlymonotone C1([0, 1]) transformations T with derivative bounded away from 1.The Perron-Frobenius operator is nowadays often defined for a general potential,hence ∂−1T can be replaced by any function from [0, 1] into (0,C] of bounded vari-ation, where C > 0, see [40], with the additional assumption that for ∂−nT (x) ≔∏n−1

j=0 ∂−1(T j(x)) for all n ≥ 2 the inequality lim sup ∥∂−nT∥1/n∞ < 1 holds, [48], [40,

I.(a)]. Here this is satisfied, as ∥∂−1T∥∞ ≤ 1/d0 < 1.If the potential is the derivative, the Perron-Frobenius operator is also referred toas Ruelle operator. The Ruelle operator also be defined with other setting, e.g.such as I may not even assumed to be finite as in [51].

The theorem we state now can be seen as a refined version of Theorem 6.4.1for this setting. It is a variation of the theorem of Ionescu-Tulcea and Marinescu,[41], given in [40, Thm. 1], but we still assume X = [0, 1], whether Hofbauer andKeller assumed X to be a totally ordered complete set.

Theorem 6.5.7 (Ionescu-Tulcea and Marinescu, [40, Thm. 1]). Let X = [0, 1],T : [0, 1]→ [0, 1] admit inverse branches and P be the Perron-Frobenius operatorof T . Then the following holds

1. P has finitely many eigenvalues λ1, . . . , λr. For all 1 ≤ i ≤ r one has |λi| = 1and exactly one eigenvalue is equal to one, which is denoted by λ1.

2. Set Ei ≔ { f ∈ L1m : P( f ) = λi f } the space of eigenfunctions for the eigen-

value λi, where 1 ≤ i ≤ r. For all 1 ≤ i ≤ r holds Ei ⊆ BV and dim(Ei) < ∞.

3. There exist projections Φi : L1m → Ei with ∥Φi∥L (L1

m) ≤ 1 and Φi ◦Φ j = 0 forall i ≠ j, where i ∈ {1, . . . , r}.

4. P =∑r

i=1 λiΦi+Ψ, whereΨ ∈ L (L1m, L

1m) withΦi◦Ψ = 0 for all i ∈ {1, . . . , r}

and supn∈N+ ∥Ψn∥L (L1

m) < ∞.

5. Ψ(BV) ⊆ BV, hence Ψ|BV is well-defined and ∥Ψn∥L (BV) ≤ Csn for someC > 0, s ∈ (0, 1).


6. Set h ≔ Φ1(1X), then h ≥ 0,∫

h dm = 1 and P(h) = h. Therefore µ ≔ hm isa T-invariant probability measure on X.

7. For any other T-invariant measure µ′ with µ′ ≪ m we have µ′ ≪ µ.

We summarise some properties of the Perron-Frobenius operator we have en-countered during the discussion in this section. For the remaining we refer to[40, 48].

Proposition 6.5.8. Let T admit inverse branches and let P, µ, h be given as inTheorem 6.5.7.

(a) For any f ∈ L1m([0, 1]), Tm( f ) = P( f ).

(b) For any f ∈ L2µ([0, 1]), Tµ( f ) = P( f · h)/h.

(c) For f ∈ L1m([0, 1]),

∫f dm =

∫P( f ) dm.

Proof. (a) Follows from the designation P( f ) = ∂m f ◦ T−1/∂m = Tm( f ).(b) Notice

1 = µ(X) =∫

1{h≠0} dhm =∫

1{h≠0} ◦ T dµ =∫

1{h≠0} ◦ T · 1{h≠0} dµ

and by an inductive argument 1 = µ(⋂

n∈N T−n{h ≠ 0}). Therefore, for all x ∈ [0, 1]where h(x) = 0 we set without loss of generality P( f ·h)

h (x) ≔ 0. With that for allf , g ∈ L2

µ([0, 1])∫g ·

P( f h)h

dµ =∫

g1{h≠0} ·P( f h)

hdµ

=

∫g1{h≠0} ·

P( f h)h

dhm

=

∫g1{h≠0} · P( f h) dm

=

∫g ◦ T1{h≠0} ◦ T · f · h dm

=

∫g ◦ T1{h≠0} · f dµ

=

∫g ◦ T · f dµ =

∫g · Tµ( f ) dµ,

since UT is dual to Tµ in L2µ([0, 1]). For an alternative approach, see [40, Lem.

8(i)].(c) By using (a) one has

∫f dm =

∫1 ◦ T f dm =

∫P( f ) dm for any f ∈

L2m([0, 1]). For another proof see [40, Lem. 6(i)]. �


Remark 6.5.9. We have shown in Proposition 6.5.8 that the dual of the Perron-Frobenius operator is the Koopman-operator.

Lemma 6.5.10. Let T admit inverse branches and let P, µ, h be given as in Theo-rem 6.5.7.

(a) The measure theoretical dynamical system ([0, 1],B,T, µ) is weakly mixingif and only if P = Φ1 + Ψ. In this case Φ1( f ) = h ·

∫f dm for all f ∈ BV.

(b) The measure µ is the only T-invariant ergodic measure, which is absolutelycontinuous to the Lebesgue measure.

Proof. (a) From Proposition 6.5.8 (a), UT ( f ) = T ∗µ( f ) and Tµ( f ) = 1/hP(h · f ),where 1/hP(h · f )(x) = 0 for all x ∈ [0, 1] such that h(x) = 0 and h ∈ BV.By assumption UT is an isometry and hence ([0, 1],B,T, µ) is weakly mixing ifand only if the spectrum σL (UT ) = {1} and all eigenfunctions are constant, seeLemma 6.5.3. These properties are shared by U∗T , [86, Satz IV.1.2], where bothUT and U∗T are considered on L2

µ([0, 1]). Note that U∗T = (Tµ)|L2µ= (P(h·)/h)|L2

µ

have eigenvalue 1 and share the same eigenfunctions. As all eigenfunctions ofP are in BV, by Theorem 6.5.7 (2) and hence BV ⊆ L2

µ([0, 1]), we have for anyeigenfunction f of P with eigenvalue λ that λ f h = P(( f /h)h) = U∗T ( f /h)h and as1 is the only eigenvalue of U∗T and its eigenfunctions are constant λ = 1, f = chµ-a.e. for some c ∈ C. Therefore σL (P) = {1} is the only eigenvalue of P andall eigenfunctions of P are spanned by h, see Theorem 6.5.7 (6). Furthermore itholds r = 1. Next it will be shown that Φ1( f ) = h ·

∫f dm for all f ∈ BV. For all

n ∈ N+ by Theorem 6.5.7 (3,4) Pn = Φ1 +Ψn. An application of Proposition 6.5.8

(c) for n→ ∞ yields∫f dm =

∫Pn( f ) dm =

∫Φ1( f ) + Ψn( f ) dm −→

∫Φ1( f ) dm + 0, (6.13)

where we made use of ∥Ψn( f )∥BV → 0 for n → ∞ from Theorem 6.5.7 (5). Nextwe use h = P(h) = Φ1(h)+Ψ(h) = Φ1(h) which follows from Theorem 6.5.7 (4,6)and deduce that E1 is spanned by h. To see this we use that a function f is aneigenfunction of P if and only if f /h is constant. Hence Φ1 : BV→ E1 = span(h)and from (6.13) we have

∫f dm =

∫Φ1( f ) dm =

∫hc dm for some constant c.

The statement then follows from∫

h dm = 1.(b) Assume there exists another T -invariant measure ν ≪ m. Then ν ≪ µ by

Theorem 6.5.7 (7) and hence ν is ergodic, as for all A ∈ B with µ(A) = 0 alsoν(A) = 0 and if µ(A) = 1 then µ(Ac) = 0, hence ν(Ac) = 0. Thus ν is ergodicand by Birkhoffs ergodic theorem, ν(A) = limN→∞ 1/N

∑N−1n=0 1A(T n(x)) = µ(A) for

µ-almost every x ∈ [0, 1]. �

There is a link between the eigenspaces of P and the tail-σ-algebra.


Lemma 6.5.11 ([40, Thm. 3(i)]). With the assumptions and notations of Theo-rem 6.5.7: { f ∈ L2

µ : f is BN-measurable} =⋂

n∈N T n(L2µ([0, 1])) =

∑ri=1 Ei. In

particular, BN is finite up to sets of µ-measure zero.

The proof of this lemma uses (6.13) and then approximates any function inL2µ([0, 1]) by functions in BV. The next proposition is a nice follow-up result to

Lemma 6.5.10. It shows that all types of mixing, in case of the Perron-Frobeniusoperator, fall together.

Proposition 6.5.12 ([40, Theorem 3.(ii)]). Let ([0, 1],B,T, µ) be weakly mixing,then ([0, 1],B,T, µ) is exact, where µ is chosen as in Theorem 6.5.7.

Proof. By Lemma 6.5.10 ([0, 1],B,T, µ) is weakly mixing. The exactness is nowimmediate with Lemma 6.5.11, as { f ∈ L2

µ : f is BN-measurable} = E1 = span(h).�

6.5.1 Spectrum for weakly mixing systemsThe results obtained here and in Section 6.3 have been appeared in [50].

The following theorem is mainly a combination of the theory introduced inSections 6.3 and 6.5. The notions for the Perron-Frobenius operator introducedin Theorem 6.5.7 and for return time combs will be used throughout this part. Toprevent confusion it is pointed out that µ = hm is here defined with respect to thePerron-Frobenius operator and does not denote a return time comb.

Theorem 6.5.13. Let ([0, 1],B,T, µ) be weakly mixing, where µ = hm, and Tadmits inverse branches. Further let f1, f2 ∈ BV be real valued. Further denoteby ηi the fi-weighted return time comb with respect to T and with reference pointy for i ∈ {1, 2}. Then for µ-almost every y the spectral return measure ˆη1 ~ ˜η2 ofη1 ~ ˜η2 is given by

ˆη1 ~ ˜η2 =

∫f1 dµ

∫f2 dµ δ0 + gm.

Here g(x) ≔∑

z∈Z cz e2πixz, where cz ≔∫

f1 · Ψ|z|( f2h) dm, for z > 0, cz ≔

∫f2 ·

Ψ|z|( f1h) dm, for z < 0 and c0 ≔∫

f1 f2 dµ−∫

f1 dµ∫

f2 dµ is an analytic function.

Proof. For n ∈ N we observe the following chain of equalities.

η1 ~ ˜η2(n) =∫

f1 ◦ T n · f2 dµ

=

∫f1 ◦ T n · f2h dm


=

∫f1 · Pn( f2h) dm

=

∫f1 · (Φ1( f2h) + Ψn( f2h)) dm

=

∫f1 ·

(h∫

f2h dm + Ψn( f2h))

dm

=

∫f1 · h dm

∫f2 · h dm +

∫f1 · Ψ

n( f2h) dm

and by definition cn =∫

f1 ·Ψn( f2h) dm. This, in tandem with Corollary 6.3.5 and

an analogous argument for n < 0, allows us to write the autocorrelation η1 ~ ˜η2 as

η1 ~ ˜η2 =

∫f1 dµ

∫f2 dµ δz +

∑z∈Z

czδz. (6.14)

Here

|cz| ≤ max{∥ f1∥∞∥ f2h∥BVC s|z|, ∥ f2∥∞∥ f1h∥BVC s|z|

}, (6.15)

where we used that BV ⊆ L∞m and that the multiplication of two functions withbounded variation is again of bounded variation. For ϕ = ϕ1 ∗ ˜ϕ1, where ϕ1 ∈

Cc([0, 1)), we have that

⟨γy,ˆϕ⟩ =∫f1 dµ

∫f2 dµ

∑z∈Z

ˆϕ(z) +∑z∈Z

cz ˆϕ(z)

=

∫f1 dµ

∫f2 dµ

∑z∈Z

ˆϕ(z) e−2πi0z +∑z∈Z

cz

∫ϕ(x) e−2πixz dm(x)

=

∫f1 dµ

∫f2 dµ ϕ(1) +

∫ϕ(x)

∑z∈Z

cz e−2πixz dm(x)

=

⟨∫f1 dµ

∫f2 dµ δ0 + gm, ϕ

⟩.

To split the series in the first equality we require that both series on the right handside are absolutely convergent. This is true for the first series, since ˆϕ(z) = |ˆϕ1(z)|2

and thus ∥ˆϕ∥1 = ∑z∈Z ˆϕ(z) =

∑z∈Z ˆϕ(z)(1, z) = ϕ(1) < ∞. To see that the

second series is absolutely convergent, notice (cz)z∈N and (c−z)z∈N are sequencesof exponential decay, see (6.15), and ϕ is a continuous function on a compactspace. With this at hand, we note that the integral and the sum of the secondcomponent in the second equality can be interchanged by Lebesgue’s dominatedconvergence theorem. The representation of g implies analytic and thus continuityof the function. �


Remark 6.5.14. If the return time combs are equal, i.e. η1 = η2, then g is a realanalytic function. Contrary to that, different return time combs do not need todepend on the same reference point y, as the proof of 6.5.13 works in the sameway, if the different reference point are chosen. The statement then holds for µ2

almost every pair of reference points.

x

y

112

112

Figure 6.2: The densities gk for k ∈ {3, 5, 10, 30, 50} of the spectral return measureof ηy, as determined in Example 6.5.15, are approximating the constant densityof height 1/12. This indicates that the decay of correlation for the observable fdecays faster for larger values of k ∈ N≥2. Here the the axis are scaled 1 : 2 andthe graph of g50 is not dashed.

Example 6.5.15. Let for k ∈ N≥2 Tk : [0, 1) → [0, 1) be given by Tk(x) = kx mod1 and set f (x) = x for all x ∈ [0, 1). Note that here the one-point compactificationis considered for [0, 1). An elementary calculation shows that Tk is a piecewisemonotonic transformation. By (8.2), m|[0,1) is an invariant measure and by [67,Thm. 2] weakly mixing. Hence for n ∈ N,

Ξ(Tk, η)(n) =∫

f · f ◦ T nk dm|[0,1)

=

kn−1∑m=0

∫ (m+1)k−n

mk−nx(knx − m) dx =

k−2n

6

kn−1∑m=0

(3m + 2) =14

kn + 1/3kn .

Using Theorem 6.3.2 and (6.14), this gives, for z ∈ Z, z ≠ 0,

Ξ(Tk, η)(z) =14

k|z| + 1/3k|z|

=14+

∫Ψ|z|( f ) · f dm|[0,1].

Hence,

γy =14

∑z∈Z\{0}

δz +∑

z∈Z\{0}

14

(k|z| + 1/3

k|z|− 1

)δz +

13δ0 =

14

∑z∈Z

δz +1

12

∑z∈Z

k−|z|δz.

6.6. SUBSTITUTION DYNAMICAL SYSTEMS 111

Combining this with Theorem 6.5.13, yields ˆγy = δ1/4 + gkm, where

gk(x) =1

12

∑z∈Z

k−|z| e−2πixz =1

12

(1

1 − 1/ke2πix +1

1 − 1/ke2πix − 1)

=1

12

(2 − 2/k cos(2πx)

1 + k−2 − 2/k cos(2πx)− 1

)=

112

k − k−1

k + k−1 − 2 cos(2πx). (6.16)

See Figure 6.2 for the graph of gk for different values of k.

6.6 Substitution dynamical systemsAs before S denotes the left shift and Σ a finite alphabet. For any S -invariantclosed X ⊆ ΣN the triple (X,B, S ) is a dynamical system, where B is generatedfrom the topology generated from the base consisting of all cylinder sets. Any mapΣ → Σ∗\{ε} is called substitution, it can be extended on Σ∗ or ΣN as a semigrouphomomorphism and is usually considered on a subset of either of theses spaceswhich will be done without further notice. Within this work, we will solely focuson primitive substitutions, which are not permutations of Σ with the exception ofthe flip-map θ on {0, 1}.

Definition 6.6.1. A substitution ζ : Σ → Σ∗ is called primitive if for all a, b ∈ Σthere exists an k ∈ N such that b is a letter of ζk(a) and |ζ(c)| ≥ 2 for some c ∈ Σ.It is of constant length, if there exists a q ∈ N+ for all a ∈ Σ such that |ζ(a)| = q.

If Σ is identified with the set {0, . . . , s} for some s ∈ N, the occurrences ofletters in a substitution ζ can be expressed by a so called incidence matrix M(ζ) ≔(|ζ(b)|a)0≤a,b≤s. For a primitive substitution ζ and any a ∈ Σ there will be a k ∈ Nsuch that |ζk(a)| ≥ 2 and hence limn→∞ |ζ

n(a)| = ∞. Further ζk(a)0 = a for somek ∈ N, a ∈ Σ and ζ2k(a) has ζk(a) as a prefix. This is an inductive argument, soζ(n−1)k(a) is a prefix of ζnk(a) and hence u ≔ limn→∞ ζ

nk(a) is a fixed point of ζk,or equivalently a periodic point of ζ. Now, for all b ∈ Σ exists an l ∈ N suchthat S lζk(a)0 = b (by a possible reassignment of k to one of its multiples). In thisway every fix point gained from b can be approached from a, respectively u viaS . As a consequence for any ζ-periodic point u, the subshift Xζ ≔ {S n(u) : n ∈ N}is independent of u. Therefore one also sees that every u ∈ Xζ has arbitrary highiterations of ζ on any letter as its factors, which leads to minimality of Xζ as thelanguage can be made up from these factors.

Definition 6.6.2. Let ζ be a primitive substitution. If a point u ∈ Xζ is a periodicpoint of ζ with period k ∈ N it is a fixed point of ζk and u is called fixed point of ζby a reassignment of ζ to the primitive substitution ζ′ ≔ ζk.


This assignment is consistent by the following proposition.

Proposition 6.6.3. For any primitive substitution ζ and periodic point u of ζ, thefollowing assertions hold.

• ({S m(u) : m ∈ N},B, S ) = (Xζ ,B, S ) = (Xζm ,B, S ) for any m ∈ N,.

• The topological dynamical system (Xζ ,B, S ) is minimal.

Proof. A proof is given by the discussion in front of the proposition, while thestatements can also be found in [71, Prp. 5.3, 5.4]. �

In [71, Ch. 5] non-primitive substitutions ζ are introduced and it is deducedthat (Xζ ,B, S ) is minimal if and only if ζ is primitive, [71, Prp. 5.5]. The defini-tion of minimality coincides with the one given in Definition 2.2.7, see [70, Prp.5.1.13]. An S -invariant probability measure of (X,B, S ) may for any u ∈ X begiven by v-limN→∞ 1/N

∑N−1n=0 δS n(u). In the case of primitive ζ, the measure the-

oretical dynamical system (Xζ , S , µ) is always uniquely ergodic, [71, Thm. 5.6],where µ admits the equality µ([w]) = limN→∞ |{n < N : u[n,n+|w|−1] = w}| for anyu ∈ Xζ , w ∈ L (u).

Proposition 6.6.4. Let ζ be a primitive substitution. The following holds

1. (Xζ ,B, S ) is uniquely ergodic.

2. (Xζ ,B, S ) is not strongly mixing.

3. It exists a countable set D such that S : Xζ\D→ Xζ\D is a homeomorphism.

4. The operator US : L2µ(Xζ) → L2

µ(Xζ) is an isometric isomorphism, where µdenotes the unique ergodic shift-invariant measure on Xζ .

Proof. The first point is due to the discussion preceding the proposition. Thesecond, third and fourth are [71, Thm. 6.5], [71, Prp. 5.13] and [71, Cor. 5.8]. �

Definition 6.6.5. For a primitive substitution ζ the spectrum of ζ is defined asthe spectrum of the isometric isomorphism US : L2

µ(Xζ) → L2µ(Xζ), where µ de-

notes the unique shift-invariant ergodic measure on Xζ . Further eigenvectors andeigenfunctions of US may be called eigenvectors and eigenfunctions of ζ.

By definition, the spectrum of ζ coincides with the dynamical spectrum of US

on Xζ .

Definition 6.6.6. For a primitive substitution ζ, denote ϱmax the maximal spectraltype of US : L2

µ(Xζ) → L2µ(Xζ), where µ denotes the unique shift-invariant ergodic

measure on Xζ . Then ζ is called discrete, singular and continuous, if ϱmax is dis-crete, singular to the Lebesgue measure and equivalent to the Lebesgue measure,respectively.


Remark 6.6.7. The averaged convolution for sequences on Z can be realised inthe same way as it is for functions on Z, as all objects are mutually the same.Formally, let Σ ⊆ Z be finite, for a sequence w ∈ ΣN, set w′ : Z→ Σ, w′(n) = 0 forn < 0 and w′(n) = un for n ≥ 0. Let v,w ∈ ΣN, define w ~˜v ≔ (w′δZ) ~ ˜(v′δZ),whenever the averaged convolution of w′δZ and ˜(v′δZ) exists.

In this spirit the spectral measure ϱ f (US ) is defined for f ∈ L2µ(Xζ) and the

f weighted return time comb µv =∑

n∈N f (S nv) δn with reference point v ∈ Xζ

is defined for bounded µ-measurable f : Xζ → C. If Σ ⊆ C, there is a canonicalchoice.

Definition 6.6.8. Let Σ ⊆ C, ζ be a primitive substitution and u be a fixed point ofζ. Define f : Xζ → Σ by v ↦→ v0. The spectral return measure ˆγu of an autocorre-lation γu = u ~˜u will be called return spectrum of ζ.

Note that in case of Definition 6.6.8 the function f is continuous, as f −1(a) =[a] for any a ∈ Σ and hence ˆγv exists for all v ∈ Xζ and coincides with the spectralmeasure ϱ f (US ).

Lemma 6.6.9 ([71, Prp. 6.1, Cor. 6.1]). Let ζ be a primitive substitution ofconstant length q and u be a fixed point of ζ. An eigenvalue λ of ζ is called contin-uous, if its associated eigenfunction is continuous. Every continuous eigenvalueis a root of unity and for λp = e2πi/p for some p ∈ N with gcd(p, q) = 1 the numberp divides every element of H(u) ≔ {n ∈ N+ : ua = u0}.

In the following it will be investigated how regular letters in a fixed point ufrom a primitive substitution ζ of constant length q are distributed. The techniquespresented can be found in [25, 71, Rem. 9, Def. 6.1], while it has been been firstmentioned in [61, 4.06]. For any k ∈ N define

S k ≔ {n ≥ 1 : uk+n = uk} , gk ≔ gcd(S k).

Fix l, k ∈ N, then there are finitely many a1, . . . , am ∈ S l that fulfil the property gl =

gcd({a1, . . . , am}) and, as ζ is primitive, there also exist n,N ∈ N with (ζN(ul))n =

uk. The equality ζ(u) = u from u being a fixed point of ζ yields by definitionlqN + n ∈ S k and (l+ ai)qN + n ∈ S k for all i ∈ {1, . . . ,m}. Hence, aiqN ∈ (S k − S k)for all i ∈ {1, . . . ,m}, this yields gcd(S k − S k)| gcd({a1qN , . . . , amqN}) = qNgl,(here “ | ” denotes the divides-symbol). Next we want to note that in generalgcd(A)|gcd(A − A) for any A ⊆ Z and hence

gk| gcd(S k − S k)| gcd({a1qN , . . . , amqN}) = qNgl, (6.17)

for arbitrary k, l ∈ N. Further note that the set S k − S k describes all factors of uof the form ukwuk for some w ∈ L (u) and by minimality the languages generated


from different fixed points u, v of ζ are the same. Hence, S k(u) − S k(u) = S l(v) −S l(v), where uk = vl for some k, l ∈ N. With that in mind the relation (6.17)is independent of the chosen fixed point and k, l if we consider all p ∈ N+ withgcd(p, q) = 1 in the sense that

{p ∈ N+ : p|gk, gcd(p, q) = 1} = {p ∈ N+ : p|gl, gcd(p, q) = 1}

for all k, l ∈ N. This ensures the following is well-defined.

Definition 6.6.10. The height of a primitive substitution ζ of length q is definedby

h(ζ) ≔ max{p ∈ N+ : p| gcd(S 0), gcd(p, q) = 1

},

where S 0 ≔ {m ∈ N+ : um = u0}. Here u ∈ Xζ is a fixed point of ζ and h(ζ) isindependent of the choice of u.

Remark 6.6.11. We would like to note that Dekking made his definition for all u,[25, Def. 8] and Queffelec states that all continuous eigenvalues of ζ are roots ofunity, hence e2πi/p for some p ∈ N. If additionally gcd(p, q) = 1, then p|S 0, hencep ∈

{p′ ∈ N+ : p′| gcd(S 0), gcd(p′, q) = 1

}, see [71, Cor. 6.1].

Proposition 6.6.12. Let ζ be a primitive substitution of constant length q over thealphabet Σ and u be a fixed point of ζ.

• 1 ≤ h(ζ) ≤ |Σ|.

• If h(ζ) = |Σ|, then u is periodic.

Proof. h(ζ) is sensitive on all factors of u of the form ukwuk for some w ∈ L (u),k ∈ N as shown above (inbetween (6.17) and Definition 6.6.10). In general, aword that maximizes the distance between the occurences of all of its letters isa periodic concatenation of a permutation of Σ, hence its height is |Σ| and it isperiodic. Another proof can be found in [71]. �

We like to finish this section with two important theorems regarding the spec-trum of primitive substitutions.

Theorem 6.6.13 ([71, Thm. 6.1]). For a primitive substitution ζ of constant lengthq, let Vq ≔ {e2πik/qn

: n ∈ N, k ∈ Z}, Wh(ζ) ≔ {λ ∈ [0, 1) : λh(ζ) = 1} and denote byG ⊆ [0, 1) the subgroup of all continuous eigenvalues of ζ. Then

G = Vq ×Wh(ζ).

The next definition will find use in the last theorem of this section, whichpresents a criterion when a subshift has discrete spectrum.


Definition 6.6.14. A primitive substitution ζ of constant length admits a coinci-dence (of order n at position position k ∈ N for the letter b), if there exists n, k ∈ Nand b ∈ Σ such that b = (ζn(a))k for all a ∈ Σ.

Example 6.6.15. The substitution ρ on Σ = {0, 1, 2} given by

ρ :

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩0 ↦→ 01211 ↦→ 01022 ↦→ 1201

and by ρ2 :

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩0 ↦→ 01210102120101021 ↦→ 01210102012112012 ↦→ 0102120101210102

one sees that ρ admits no coincidence of order n = 1, but it admits coincidencesof order n = 2, for i ∈ {0, 1, 6, 11, 14}. For the Thue-Morse substitution τn

TM(0) isthe flip of τn

TM(1) for all n ∈ N and hence it does not admit a coincidence.

Theorem 6.6.16 ([25]). The maximal spectral type of a primitive substitution ζ ofconstant length with height h(ζ) = 1 is discrete, if and only if ζ admits a coinci-dence.

Remark 6.6.17. Substitutions with height one are also said to be pure. While thetheorem has been first proven by Dekking, [25], it is stated in a more applicableform in [70, Prp. 5.5.8], but without a proof. Another proof of Theorem 6.6.16 ispresented in [71, Ch. 6.1, Thm. 6.6].

Proposition 6.6.18. For a primitive aperiodic substitution ζ of constant length qwith height h(ζ) = 1 is the maximal spectral type, ϱmax of S , a discrete measureand non-zero on the set {q−nZqn : n ∈ N} ⊆ [0, 1].

Proof. That ϱmax(S ) is a discrete measure is due to Theorem 6.6.16, set Vq ≔{x ∈ [0, 1] : ϱmax(S )(x) ≠ 0}. It is known that ϱmax(S )(x) ≠ 0 is equivalent toe2πix being an eigenvalue of S on L2

µ(Xζ ,C), see Proposition 6.4.4 4 or [71, Prp.2.14], where (Xζ ,B, S , µ) is as in Proposition 6.6.4. In particular all eigenvaluesare continuous eigenvalues by [71, Thm. 6.2] and with Theorem 6.6.13 we obtainVq = {q−nZqn : n ∈ N} as mentioned in the theorem. �

Remark 6.6.19. Let u be a fixed point of a substitution ζ on Σ = {0, 1}. Animmediate answer for an approximation of the return spectrum of ζ is then givenvia (6.3), by

ˆγu(x) = v-limN→∞1

N + 1

N∑

n=0

une2πinx

2 m[0,1]

where m[0,1] denotes the Lebesgue measure. The limit exists, as (Xζ ,B, S , ) isuniquely ergodic and an 1[1]-weighted return time comb with reference point u


is considered, where 1[1] is continuous. This well known approximation of theBochner transform for autocorrelations can for example be found in [71, Ch.4.3.3, Prp. 4.11(b)], where it is stated for primitive substitutions. It appears in thecontext of continuous eigenvalues, see Lemma 6.6.9, and leads to a description ofthe maximal spectral type in [71, Ch. 7.1.2]. Note that this approximation workslikewise with the Thue-Morse substitution (see Section 8.5.1 for an introduction),therefore it does not indicate if the spectral measure has a discrete, continuous orsingular part.

Chapter 7

Quasicrystals

This work has a focus on aperiodic subshifts, which can be considered as qua-sicrystals on Z ⊆ R. However, in Proposition 7.1.1 we will see a strong connec-tion between quasicrystals on Z and spectral measures. Further, in Section 6.3, wedefine a Dirac comb η for which the general procedure of deriving the diffractionof η will be well-defined only almost surely. For the sake of giving an answerof what is here understood as the general procedure for deriving the diffraction,a rough explanation is given in the following paragraphs. The notions and defi-nitions presented there are therefore vague by nature and hence will not be usedoutside this chapter.

A quasicrystal can be understood as an aperiodic subset of a discrete countablesubgroup L of a locally compact abelian group G. It is often assumed that L is co-compact and is then referred to as lattice. If one places weights on each elementof L, this defines a Dirac comb µ on G (complex or non-negative). These µ thenexperience some sort of averaged convolution and we call the result γ, where itexists. The existence of γ can often be ensured by using some sort of ErgodicTheorem and is generally referred to as autocorrelation. This generally ensuresthat γ is Fourier transformable in the sense of [3]; for non-negative measures onemay also consider [13]. This is often done by using some sort of abbreviation ofthe Poisson summation formula, designed to fit the lattice L and autocorrelationγ defined in the context of the authors. As supp(γ) will always be a discretecountable subset of L we like to remark that instead of relying on the theory givenin [3, 13], it is often enough to use Bochner’s theorem. The Fourier transformˆγ ofγ derived along these lines is called diffraction. Special attention is put to discreteˆγ, which is called pure point in this context.

Next, some works are mentioned, which execute this sloppy introduction in arigorous way and I apologise in advance for all the important works in this fieldwhich I have not had the time to properly study to pay tribute to. All of them intro-duce many tools in much generality which are also used in this work, altough often

117

118 CHAPTER 7. QUASICRYSTALS

slightly different or tailored to the current needs. One of these is [59], which isabout weakly almost periodic measures for group actions. There the authors showthat every such measure has a unique decomposition into a pure point diffractivepart and continuous diffractive part. But due to the rather abstract definition ofweak almost periodicity, a decomposition for explicit measures is generally hardto obtain. In Section 6.5.1 it is shown that weighted return time combs for mixingtransformations are, in general, not weakly almost periodic. Other works dis-cussing the effect of mixing conditions on the autocorrelation of lattices include[12, 64, 82]. For autocorrelations and diffraction stemming from group actionswe also like to mention [57, 65, 73]. Especially in [8], Baake and Lenz use thework of Gouere [34] to construct an autocorrelation, which takes the average ofall ergodic measures on the space of translation bounded measures under groupactions. With this setting they are able to forego the almost-sure existence of areference point we assume for return time combs with the price of blurring theirresult as all ergodic measures are taken into account. This is used in [8, Thm. 7]to give a link between diffraction and dynamical spectrum. The approach of av-eraging with measures is also used in [58] to define an autocorrelation for unitarytransformations, [58, Lem. 4.1, Def. 4.2]. In Theorem 5.3 the authors identifytheir diffraction with spectral measures and embark on the pure point case in The-orem 5.5. Weighted Dirac combs supported on locally compact abelian groupswith a focus on pure point diffraction have been studied in [7, 9, 74]. Special at-tention in this direction should be paid to regular model sets and cut and projectschemes, which have been extensively studied and shown to be pure point, see forinstance [6, 7, 72, 75, 74] and references therein.

7.1 Cut and Project SchemesThis section will give a short introduction to cut and project schemes as defined ine.g. [74, 6]. Here we assume that G is additionally second countable. As above, alattice L ⊆ G is a discrete, co-compact subgroup and mL is a Haar measure on L.With these assumptions the Fourier transform ˆmL of mL exists and has support inˆG/L such that the Poisson summation formula

⟨mL, f ⟩ = ⟨ˆmL, f ∨⟩

holds for f ∈ {Cc(ˆG) : ˆf ∈ L1(G)}. The set ˆG/L is again a lattice and isomorphicto the annihilator of L, which is given by L0 ≔ {χ ∈ ˆG : ∀x ∈ L, χ(x) = 1}.

For G = Rd this also has a matrix counterpart, as there is always a matrixM such that L = MZd. By definition ˆRd = {y : Rd → [0, 1) : y is a continuoushomomorphism} ≅ Rd in virtue of x ↦→ e2πiyT x, y ∈ Rd. With that

L0 ≅{y ∈ Rd : ∀x ∈ L, ⟨x, y⟩ = 1}

7.1. CUT AND PROJECT SCHEMES 119

={y ∈ Rd : ∀x ∈ Zd, e2πiyT Mx = 1}

={y ∈ Rd : ∀x ∈ Zd, yT Mx ∈ Z} = (M−1)TZd.

A Cut and Project Scheme is a triple (G,H, L), where G,H are locally compactabelian second countable groups and L ⊆ G × H is a lattice such that for thecanonical projections πG : G × H → G, πH : G × H → H, we have πG |L is ainjection and πH(L) is dense in H. One can define a functional

η ≔∑

(x,y)∈L

f (y)δx

on the cut and project scheme, where f : H → C is a bounded function. In [74]the authors further assumed f to be Riemann integrable (for Riemann integrablefunctions on Banach spaces, see e.g. [2]) and then concluded that the autocorrela-tion γ with respect to any Van-Hove sequence (see e.g. [80]) exists and is given byγ = dens(L)

∑(x,y)∈L( f ∗ ˜f )(y)δx, where dens(L) is a constant, [74, Prp. 5.1]. They

also calculated the diffraction ˆγ = dens(L)2 ∑(x,y)∈L0

| f ∨|(y)δx, [74, Thm. 5.2(iii)].

7.1.1 Rotation as Cut and Project SchemeLet α ∈ R be irrational. The CPS is given by (G,H, L), where G = Z, H = [0, 1)and L = {(n, {nα}) : n ∈ Z}. Then (Z × [0, 1))/L ≕ K ≅ [0, 1), as all elementsin {(0, x) : x ∈ [0, 1/2)} belong to different equivalent classes. Note that theelements in {0} × (1/2, 1) are already covered by the aforementioned set. As themapping F : (m, y) ↦→ (m+ 1, y+ α) is a continuous bijection, a calculation showsFn(0, x) = (n, {x + nα}) and (n, {x + nα}) − (0, x) = (n, {nα}), hence [0, 1/2] ≅ K ∋[x] = {Fn(0, x), Fn(0,−x) : n ∈ Z}. We know ˆZ × [0, 1) ≅ [0, 1) × Z, where bydefinition L∗ ≅ {χ : Z× [0, 1)→ [0, 1) cont. hom. : χ(L) = {1}}. With this we have

χy,m((n, {nα})) =e2πiny · e2πinαm = 1 ⇐⇒ 1 = e2πin(y+mα) =⇒ y = −mα,

hence L∗ = {({−mα},m) : m ∈ Z} ⊆ [0, 1) × Z. With that at hand we deduce[0, 1) × Z/L∗ ≅ [0, 1/2] × {0} ≅ ˆK.

Note that {(n, nα) : n ∈ Z} ↪→ R × R is not a lattice, as it is not co-compact.But {(n, {nα}) : n ∈ Z} ≕ L is a lattice and R × [0, 1)/L ≅ [0, 1) × [0, 1/2], byFw : (v, y) ↦→ (v + w, {y + wα}). Additionally ˆR × [0, 1) ≅ R × Z and L∗ ≅ {(l −mα,m) : l,m ∈ Z}, as the direct products ⟨(−mα,m), (n, nα)⟩ = −mnα + mnα = 0and ⟨(l, 0), (n, nα)⟩ = ln ∈ Z. One may also consider the equivalent direct productx • y ↦→ e2πi⟨x,y⟩ and then ask when x • y = 1, which is more close to the originaldefinition. We get (R × [0, 1))/L∗ ≅ [0, 1) × {0}, as (y,m) ↦→ (y − mα,m + 1) is abijection and hence for each m the elements in R × {m} are object to the relationby Z × {m}.

120 CHAPTER 7. QUASICRYSTALS

For the embedding of a Dirac comb η on Z to R, its diffraction will be aperiodic case of its original diffraction. If, for instance, f : R→ C positive definitefunction and η ≔ f δZ is a functional on R with support in Z, then, by ˆη = ˆf δZ =ˆf ∗ ˆδZ = ˆf ∗ δZ, one can also consider η to be a measure on Z in terms of itsFourier transform. As the previous calculation exhibited some features, like usingthe Bochner transform, Theorem 5.2.2, for f as a function on R restricted on Z,the following proposition presents a more detailed proof and makes use of thetechniques introduced in Sections 5.2 and 5.2.1.

Proposition 7.1.1. Let η ∈ SFP(Z) ⊆ C′c(Z) and denote the canonical embeddingof η into Cc(R)′ by ηR, then

⟨ηR, ϕ⟩ = ⟨ˆη ∗ δZ, ϕ∨⟩,for all ϕ ≔ ϕ1 ∗ ˜ϕ1, where ϕ1 ∈ Cc(R).

For a definition of ˆη ∗ δZ see Appendix C.1.

Proof. Note that ηR is a linear combination of positive definite functionals on Rand hence transformable. To see this consider a positive definite η′ ∈ CP(Z), then⟨η′R, f ⟩ = ⟨η′, f|Z⟩ and if f is positive definite, then f|Z is also positive definite, asthe definition of positive definiteness is still satisfied for a subset Z of R. Hence⟨η′, f|Z⟩ ≥ 0, which yields that η′R is positive definite.

The Bochner transform of η will be denoted by ˆη and for any ϕ = ϕ1 ∗ ˜ϕ1,where ϕ1 ∈ Cc(R) we have

⟨ηR, ϕ⟩ =∑z∈Z

η(z)ϕ(z) =∑z∈Z

∫ 1

0e2πizx dˆη(x)ϕ(z) =

∫ 1

0

∑z∈Z

ϕ(z)e2πizx dˆη(x).

Summation and integration could be interchanged, as ϕ has continuous supportand therefore the sum has finitely only many arguments. As ϕ ∈ L1

m(R) ∩ L2m(R)

we have ˆϕ ∈ L1m(R), by a theorem of Plancherel and hence (ϕ)∨ exists. Thus

ϕ(z)e2πizx =

∫R

ˆϕ(v)e2πivze2πizx dm(v)

=

∫R

ˆϕ(v)e2πi(v−x)z dm(v) =∫R

ψx(w)e−2πiw(−z) dm(w) = ˆψx(−z),

where ψx(w) ≔ ˆϕ(w − x) ∈ Cc(R,C), we have

⟨ηR, ϕ⟩ =

∫ 1

0

∑z∈Z

ˆψx(−z) dˆη(x)

=

∫ 1

0

∑z∈Z

ψx(−z) dˆη(x) =∫ 1

0

∑z∈Z

ˆϕ(−x − z) dˆη(x) = ⟨ˆη ∗ δZ, ϕ∨⟩,by a use of the Poisson summation formula, see Proposition C.3.3. �

Chapter 8

β-transformations

Palmer studied in her Phd thesis, [66], transformations

T ≔ Tβ,α : [0, 1]→ [0, 1], x ↦→ βx + α mod 1 = {βx + α} ,

for x ≠ 1 and T (1) ≔ limx↗1 T (x), where β > 1, α > 0 ∈ R+ with respect totransitivity, weak Bernoullicity and their spectral properties. If Lebesgue abso-lute continuous T -invariant measures are considered, the dynamical system withrespect to T is measure theoretically isomorphic to the one for T |[0,1), see Exam-ple A.1.2. Here, ([0, 1),+) will always considered to be a compact space withaddition modulo 1 and we will generally consider T on [0, 1) instead of [0, 1] anddo so without further notice. In fact the only section in which we stick to [0, 1] isSection 8.4 for historical reasons.

Special attention has been put to T , if their parameters are inside the set

∆ ≔{(β, α) ∈ R2 : 1 < β < 2, 0 ≤ α ≤ 2 − β

},

and every T (x) = {βx + α} is called (intermediate) β-transformation, which maybe either on [0, 1) or [0, 1] and in the latter case T (1) = β − 1 + α. Their mostdistinct feature from transformations with parameters (β, α) ∉ ∆ is that they admitexactly two branches. In Figure 8.1 one can see that indeed, due to β > 1, T isonly not injective if it maps into [T (0),T (1)).

8.1 Symbolic space for β-transformationsIn this section we briefly develop a way of connecting the dynamics of ([0, 1),T ),where T is a β-transformation, to ({0, 1}N, S ), such that for cylinder sets T ([w]T ) =[S w]T for finite words w ∈ {0, 1}∗, which will be explained in the following para-graph. For more information on symbolic spaces see Section 2.2.

121

122 CHAPTER 8. β-TRANSFORMATIONS

Any β-transformation T splits into two branches

T0 : [0, γ)→ [α, 1), x ↦→ βx + α, T1 : [γ, 1)→ [0, β − 1 + α), x ↦→ βx + α − 1,

where γ ≔ (1 − α)β generally denotes the T-discontinuity of a β-transformation.Note that the discontinuity is always at 0 for T : [0, 1) → [0, 1) and not at γ, butT−1(0) = γ.

1

10

0

T (0)

T (1)

γ

Figure 8.1: T : x ↦→ {βx + α} for β = 1.42, α = 0.102.

Following the idea of the coding of a point, we look at iterations of T−1 start-ing with T−1([0, 1)) = T−1

0 ([0, 1)) ⊎ T−11 ([0, 1)), which encompass all possible

codings with respect to T−10 and T−1

1 . Therefore we identify for every n ∈ N+each word w ∈ {0, 1}n with the n-cylinder [w]T given by the set T−1

w ([0, 1)) ≔T−1

w1◦ . . . ◦ T−1

wn([0, 1)), as long as the set is not empty. This let us define the lan-

guage L(T ) ≔ {w ∈ {0, 1}∗ : T−1w ([0, 1)) ≠ ∅}. It is natural in this context to set

T 0(x) ≔ x for any x ∈ [0, 1), Ln(T ) ≔ L(T ) ∩ {0, 1}n and L0(T ) ≔ {ϵ}. Withthese definitions we have T ([w]T ) = [S w]T . Indeed, the lexicographical ordering(e.g. from

∑2−(1−wi)) preserves the relations of the elements in [0, 1) in the sense

that for x ∈ [w], x′ ∈ [w′] where w,w′ ∈ Ln(T ), w ≠ w′, the inequality x < x′

holds if and only if∑

2−(1−wi) <∑

2−(1−w′ i). The reason comes from the image ofT−1

0 given by [0, γ) and T−11 given by [γ, 1), which splits the [0, 1) interval into a

“left side” and a “right side”. In addition to that T n : [w]T → T n([w]T ) is a bi-jection and a Diffeomorphism on the interior of [w]T , moreover (T n)′ = βn and(T−1

w )′ = 1/βn. Further note that left/right endpoints of [w] will remain left/rightendpoints of [0w] and [1w] respectively as long as their preimages under T0 andT1 respectively are not empty, since (T−1

w )′ = 1/βn. On the other hand, the bijec-tions T−1

0 : [T (0), 1) → [0, γ) and T−11 : [0,T (1)) → [γ, 1) give us inductively that

the preimages of γ (except for 0 and 1) are the endpoints of each cylinder. To see

8.2. CATEGORISATION OF β-TRANSFORMATIONS 123

that, suppose γ1 = T−1w1

(γ), γ2 = T−1w2

(γ) such that [w1] = [γ1, γ2), then by strictmonotonicity of T−1

i for i ∈ {0, 1}, the cylinder [iw] equals one of the followingintervals

[T−1i (γ1),T−1

i (γ2)), [0,T−1i (γ2)), [T−1

i (γ1), γ), [γ,T−1i (γ2)), [T−1

i (γ1), 1). (8.1)

While limx→(1−i) T−1i (x) = γ, hence the next endpoints of the cylinders (with 0 or

1 as an endpoint) are given by γ. We have proved the following lemma

Lemma 8.1.1. The partition induced by{T− j(γ) : j ∈ {0, . . . , n − 1}

}equals the

partition given by {[w]T : w ∈ Ln(T )} and each [w]T (except the lexicographicallyfirst) has exactly one element of

{T− j(γ) : j ∈ {0, . . . , n − 1}

}as its left endpoint.

As an aside we remark that⋁n

j=0 T− j({[0, γ), [γ, 1)}) = {[w] : w ∈ Ln(T )} ≠T−n({[0, γ), [γ, 1)}), as the sets in T−n({[0, γ), [γ, 1)}) are in general collections ofintervals.

Remark 8.1.2. The derived language exhibits kneading sequences described in[33]. They determine a map up to topological conjugacy and the author used thisto establish a connection to Lorenz maps.

8.2 Categorisation of β-transformationsIn Section 8.1 we generated a language for a β-transformation from its preimages.To study these objects further it is interesting to see how often the preimage ofγ can be taken until it is not unique anymore, which is presented in the nextdefinition.

Definition 8.2.1. Let n ≥ 2, define

Cn ≔{(β, α) ∈ ∆ : min

{j ∈ N : T− j(γ) ∈ [T (0),T (1))

}= n − 2,T (x) = {βx + α}

}as a subset of ∆.

Note that the backward orbit of γ for any β-transformation T given by thesequence (T−n(γ))n∈N is only periodic if it contains T (0) at some point. That is,there exists a periodic sequence (xn)n∈N, where xn ∈ T−n(γ) for all n ∈ N. A directconsequence of Definition 8.2.1 is that for T (x) = {βx + α}, where (β, α) ∈ Cn,n ∈ N+ we have |T−m(γ)| = 1 for all 1 ≤ m < n and hence |Lm(T )| = m + 1,whereas |Ln(T )| = n + 2. Therefore the language Ln−1(T ) is Sturmian of leveln − 1, see Definition 2.2.4. The upcoming notion is now well-defined


Definition 8.2.2. Let n ≥ 2, T (x) = {βx + α}, where (β, α) ∈ Cn. The set ofdiscontinuities

{T− j(γ) : j ∈ {0, . . . , n − 1}

}of T n will be enumerated such that

d1 < d2 < . . . < dn < dn+1

and the words in Ln(T ) are enumerated such that for all j ∈ {1, . . . , n+ 1} we haved j ∈ [w j]T , where the remaining word is denoted by w0 and 0 ∈ [w0]T .

Remark 8.2.3. From T−n+2(γ) ∈ [T (0),T (1)), together with strict monotonicityof T we yield d1 = T−1

0 ◦ T−n+2(γ) and dn+1 = T−11 ◦ T−n+2(γ).

In the upcoming lemma we will determine the fix points of the map T n corre-sponding to (β, α) ∈ Cn. The proof is essentially an application of the intermediatevalue theorem for T n : [T−i(γ),T− j(γ)) → [T n−i(γ),T n− j(γ)) and the techniquescan be found in [66, p. 51, Lemma 1].

Lemma 8.2.4. Let n ≥ 2, (β, α) ∈ Cn, the corresponding map T n has exactly nfixed points, which are are denoted by z1, . . . , zn and suffice the ordering

d1 < z1 ≤ d2 < z2 ≤ . . . ≤ dn < zn ≤ dn+1.

Proof. First we show [T−i(γ),T− j(γ)) ⊆ [T n−i(γ),T n− j(γ)). By the assumptionswe have made T−(n−2)(γ) ∈ [T (0),T (1)) and for some w,w′ ∈ Ln(T ) it holds[T (0),T−(n−2)(γ)) ⊆ [w] and [T−(n−2)(γ),T (1)) ⊆ [w′]. In addition to that themap T n is a bijection on each of these intervals and by strict monotonicity, as(T n−1−i)′ = βn−1−i, we have T n−i(0) ≤ T−i(γ) and T−i(γ) < T n−i(1). Hence for anyw ∈ Ln(T ) such that there are 0 ≤ i, j ≤ n− 2 with [w] = [T−i(γ),T− j(γ)) we have[T−i(γ),T− j(γ)) ⊆ [T n−i(0),T n− j(1)).

By applying T n on [T−i(γ),T− j(γ)) we have that x↘ T−i(γ) implies T−i(γ)↘0 and conversely x ↗ T− j(γ) implies x ↗ 1. Hence T n(x) → T n−i(0) for x ↘T n−i(0) and T n(x) → T n− j(1) for x ↗ T− j(γ). By continuitiy of T n the imageof [T−i(γ),T− j(γ)) is connected and contains [T n−i(γ),T n− j(γ)). Therefore we canapply the intermediate value theorem, together with the strict monotonicity of T n

and yield that there is exactly one fixpoint.As the above holds for any w ∈ Ln(T ) such that there are 0 ≤ i, j ≤ n − 2 with

[w] = [T−i(γ),T− j(γ)), by Remark 8.2.3, the only cases which remain to checkare [w1] = [d1,T−i(γ)) and [wn] = [T− j, dn) for some fixed 0 ≤ i, j ≤ n − 2. But asd1 = T−1

0 ◦Tn−2(γ) and dn+1 = T−1

1 ◦Tn−2(γ), we have T n(d1) = 0 and T n(x)→ 1 for

x ↗ dn and hence [d1,T−i(γ)) ⊆ [0,T n−i(1)) as well as [T− j(γ), dn) ⊆ [T n− j(1), 1),which completes the proof. �

The periodic points derived from Lemma 8.2.4 will be fixed with the followingdefinition.

8.2. CATEGORISATION OF β-TRANSFORMATIONS 125

Definition 8.2.5. Let n ∈ N+, T : x ↦→ {βx + α}, where (β, α) ∈ Cn. The map T n

has exactly n fixed points (zi)ni=1, which will be enumerated such that

z1 < z2 < . . . < zn.

Lemma 8.2.6. Let n ≥ 2, (β, α) ∈ Cn, T : x ↦→ {βx + α} and set 1 ≤ k ≤ n suchthat γ ∈ [zn−k, zn−k+1). The following holds

i) γ = dn−k+1.

ii) gcd(n, k) = 1.

iii) T (zi) = z(i−1+k mod n)+1 for all i ∈ {1, . . . , n}.

iv) Applying T−1 each time we have

0 ↦→ d(−k mod n)+1 ↦→ d(−2k mod n)+1 ↦→ d(−3k mod n)+1 ↦→ . . .

↦→ d(−(n−1)k mod n)+1 = dk+1 ↦→ {d1, dn+1}

Proof. The proofs will be done in a different order

i) Holds by definition of k, together with Lemma 8.2.4. The fact 1 ≤ k ≤ nfollows from Remark 8.2.3.

iii) From i) we know T (dn−k+1) = T (γ) = 0 and as T0,T1 are strictly increasingthe T n-fixpoints T0 : zn−k ↦→ zn, T1 : zn−k+1 ↦→ z1. If not assume that there isanother T n-fixpoint T0(zn−k) < 1, which would imply zn−k < T−1

0 (z) < γ bystrict monotonicity of T0. Which is a contradiction to Lemma 8.2.4 giventhe definition of dn−k+1 = γ. By the same argument we deduce T0 : zn−k− j ↦→

zn− j for 0 ≤ j ≤ n − k + 1 and T1 : zn−k+1+ j ↦→ z1+ j for 0 ≤ j ≤ k − 1.

iv) By definition of k we have with iii) 0 ↦→ γ = dn−k+1 = d(−k mod n)+1. All othermappings are a combination of 8.2.4 and iii). While the last mapping is dueto Remark 8.2.3.

ii) Is a direct consequence of iv).

�

Lemma 8.2.7. The intervals [0, z1), [z1, z2), . . . , [zn−1, zn), [zn, 1) are a partition of[0, 1) and T restricted to each interval is a bijection. In particular

T : [0, z1)→ [T (0), zk+1), T : [zn, 1)→ [zk,T (1)),T : [zn−k, zn−k+1)→ [zn, 1) ∪ [0, z1),

T :[z j, z j+1

)→

[z( j+k mod n)+1, z( j+1+k mod n)+1

),

where j ∈ {1, . . . , n − 1}\{n − k}. Especially γ = dn−k+1 ∈ [zn−k, zn−k+1).


Proof. As T and hence T0,T1 are strictly increasing maps, we only need to checkwhere the endpoints of the intervals are mapped and this is already given byLemma 8.2.6 iii). The description of γ and the image of [zn−k, zn−k+1) is then aconsequence of Lemma 8.2.4. �

As γ ∈ [zn−k, zn−k+1) its image, under T , consists of two disjoint sets. A furtheranalysis can be done by paying attention to the case (T (0), zk+1) ∩ (zk,T (1)) =(T (0),T (1)) and hence zk < T (0) < T (1) < zk+1, which motivates the followingdefinition.

α

β

2

11

0 12

13

23

14

34

15

25

35

45

√2

3√24√25√2

D1,2

D1,3 D2,3

Figure 8.2: Plot of all Dk,n for n = 1, . . . , 5 and all viable choices of k.

Definition 8.2.8. For each (β, α) ∈ Cn let the map T : x ↦→ {βx + α} admit thedefinitions for (zi)n

i=1 and k ∈ N from Lemma 8.2.6. With these associations in

8.3. THE PARRY MEASURE 127

mind we define for each 0 < k < n, n ≥ 2 with gcd(k, n) = 1

Dk,n ≔{(β, α) ∈ Cn : zk, zk+1 ∉ (T (0),T (1))} ⊆ Cn ⊆ ∆.

Palmer and Glendinning discovered that each Dk,n ⊆ ∆ is non-empty for allk, n ∈ N+, gcd(k, n) = 1 and both authors present a description of all Dk,n withoutthe need to calculate the periodicities first. An alternative description of the setsis given in Theorem 8.2.9, while for a visualisation see Figure 8.2.

Theorem 8.2.9 ([66]). Let k, n ∈ N, (β, α) ∈ ∆, n√β and let nk, nr ∈ N be such that

n = nkk + r, where 0 ≤ nr < k. Then (β, α) ∈ Dk,n if and only if

1 − β + β∑r

j=1 W j∑ni=1 β

i ≤ α ≤βn + β − βn+1 − 1 + β

∑rj=1 W j∑n

i=1 βi .

Here

W j ≔ W j(β, k, n) ≔l j∑

i=1

β(kr−k j−1−i)nk+r− j, j ∈ {1, . . . , r}.

The remaining parameter are chosen as follows: Let k j, r j ∈ N be such that jk =k jr + r j, where 0 ≤ r j < r for all j ∈ {0, . . . , r}. At last define l j ≔ k j − k j−1 forj ∈ {1, . . . , r}.

Proof. [66, p. 2.27, Thm. 4]. Palmer has a slight mistake in (ii) of Theorem 4,which was corrected by Glendinning in [33, Prp. 2]. �

8.3 The Parry measureFor β > 1 and 0 ≤ α < 1 it has been shown by Parry that for T (x) ≔ {βx + α}for x ∈ [0, 1) and T (1) ≔ limx↗1 T (x) the dynamical system ([0, 1],B,T ) hasa T -invariant probability measure µ ≪ m, [68, Thm. 5], called Parry measureand h ≔ ∂µ/∂m is called Parry density. The assumption µ ≪ m yields that µis unique, which is immediate by Lemma 6.5.10 (b) and builds a link betweenβ-transformations and the Perron-Frobenius operator. For these particular trans-formations Parry showed in [69, Thm. 6] that µ = hm, where

h(x) ≔ Ch

∞∑n=0

β−n (1[0,T n(1))(x) − 1[0,T n(0))(x)

). (8.2)

and Ch denotes the normalizing constant such that∫

h dm = 1. Neccessarily thechoice of h in terms of Theorem 6.5.7 coincides with h in (8.2). The measure µ is


ergodic and maximal by [39, Thm. and Thm. 2], that is, its measure theoreticalentropy h(µ) = htop(T ) = log(β), which is shown in [38, Thm. 3]. A collection ofresults for invariant measures of β-transformations is given in [66] on the pages2.1ff. With that β-transformations associated with Dk,n can be further describedusing their combinatoric properties.

Lemma 8.3.1. Let (β, α) ∈ ∆ and 1 ≤ k < n with gcd(k, n) = 1. Define the mapTβ′,α′ for

β′ ≔n√β, α′ ≔

β′ − 1(β′)n − 1

⎛⎜⎜⎜⎜⎜⎜⎝−(1 − α)(1 − 1/β′) +r∑

j=1

W j(β′, k, n)

⎞⎟⎟⎟⎟⎟⎟⎠ ,where r and W j(β′, k, n) as in Theorem 8.2.9 for all j ∈ {1, . . . , r}. Set z− ≔T n−1β′,α′(0), z+ ≔ T n−1

β′,α′(1) and the define the map ι : [z−, z+) → [0, 1), x ↦→ x−z−z+−z−

.We then have (β′, α′) ∈ Dk,n and Tα,β and T n

α′,β′ |[z− ,z+)are topologically conjugate

via ι, i.e. Tα,β

ι≅ T n

α′,β′ |[z− ,z+), where the one-point compactification of [z−, z+) is

considered.

Proof. The argument is essentially given in [33] on page 408-410 using the cal-culations done by Palmer. Let (β, α) ∈ ∆ and 1 ≤ k < n with gcd(k, n) = 1.Set β′ ≔ n

√β, then the mapping T n

β′,α′ : [T n−1β′,α′(0),T n−1

β′,α′(1))→ [T n−1β′,α′(0),T n−1

β′,α′(1)) iswell-defined for any α′ ∈ (0, 2 − β′) such that (β′, α′) ∈ Dk,n, [66, p.2.21, Cor.].We define the following parameters

z− ≔T n−1β′,α′(0) = α′

n−1∑i=0

β′i −

∑rj=1 W j

β′,

z+ ≔T n−1β′,α′(1) = β′n−1 − β′n−2 + α′

n−1∑i=0

β′i −

∑rj=1 W j

β′,

α′− ≔1 − β′

(∑rj=1 W j − 1

)β′

∑n−1i=0 β

′i, α′+ ≔

β′∑r

j=1 W j − β′n+1 + β′n + β′ − 1

β′∑n−1

i=0 β′i

.

From Theorem 8.2.9 we have that (β′, α′) ∈ Dk,n if and only if α− ≤ α′ ≤ α+(Note that α′− and α′+ depend on β′) and the explicit representation for z− and z+are from [33, Eq. (19,20)], where one also has that ι : [z−, z+)→ [0, 1), x ↦→ x−z−

z+−z−.

With that we have α′ ∈ [α′−, α′+] and hence (β′, α′) ∈ Dk,n. In the following we

will check that for any α ∈ (0, 2 − β) the choice of α′ is well defined and givenas in the statement of the lemma. For that we define the β-transformation Fα′ ≔

ι ◦Tβ′,α′ ◦ ι−1 : [0, 1)→ [0, 1), by x ↦→ β′nx+α′F , where α′F ≔ 1− β′

∑rj=1 W j−α

′∑n−1

i=0 β′i

β′−1for any α− ≤ α′ ≤ α+, compare [33, Eq. (21)]. To describe the lower and upper

8.3. THE PARRY MEASURE 129

bound for α′F we choose α′ to be either α′− or α′+. With that the bounds are givenby

(α′−)F =1 − β′∑r

j=1 W j − α′−

∑n−1i=0 β

′i

β′ − 1= 1 − β′

1 − 1β′

β′ − 1= 1 −

β′

β′ − 1β′ − 1β= 0.

(α′+)F =1 − β′∑r

j=1 W j − α′+

∑n−1i=0 β

′i

β′ − 1= 1 −

β′

β′ − 1−(−β′n+1 + β′n + β′ − 1)

β′

=1 −(β′ − 1)(β′n − 1)

β′ − 1= 2 − β′n = 2 − β.

Hence α′F ∈ (0, 2 − β). Therefore it exists an α′ ∈ [α′−, α′+] s.t. Fα′ = Tβ,α, which

yields the claimed existence of a conjugacy ι. With that we choose α′F = α anddeduce from [33, Eq. (21)]

α = 1 − β′∑r

j=1 W j − α′∑n−1

i=0 β′i

β′ − 1

⇔ (1 − α)(1 − 1/β′) =r∑

j=1

W j − α′

n−1∑i=0

β′i

⇔ α′ =β′ − 1β′n − 1

⎛⎜⎜⎜⎜⎜⎜⎝−(1 − α)(1 − 1/β′) +r∑

j=1

W j(β′, k, n)

⎞⎟⎟⎟⎟⎟⎟⎠ ,which concludes the proof. �

The upcoming corollary is a summary of Lemma 8.3.1.

Corollary 8.3.2. Let (β, α) ∈ ∆ and 1 ≤ k < n with gcd(k, n) = 1. It exists( n√β, α′) ∈ Dk,n such that Tβ1/n,α′ restricted on [T n−1

β1/n,α′(0),T n−1

β1/n,α′(1)) is topologi-

cally conjugate to Tβ,α. Take note that the one-point compactification of is consid-ered on [T n−1

β1/n,α′(0),T n−1

β1/n,α′(1)).

Some properties of the conjugacy on [z−, z+) can be carried over to more inter-vals in the case zk ≤ T (0) < T (1) ≤ zk+1, which is fulfilled for every T associatedwith Dk,n, where (zi)n

i=1 let us observe T as a bijective mapping, 8.2.7. This isspecified within the following lemma.

Lemma 8.3.3. Let (β, α) ∈ Dk,n. Then the map T = Tβ,α : x ↦→ {βx + α} has n do-mains Ii = [T i(0),T i(1)), i ∈ {1, . . . , n−1} and In = [0,T n(1))∪[T n(0), 1) on whichT n : Ii → Ii, i ∈ {1, . . . , n} is measure theoretically isomorphic to Tβn,α′ : x ↦→{βnx, α′} with (βn, α′) ∈ ∆ up to a constant. If ι is as in Lemma 8.3.1, one can setα′ ≔ ι(T n−1(0)) and the following diagram commutes


Ii Ii

[0, 1) [0, 1)

T nβ,α

ι ◦ T n−1−iβ,α

Tβn,α′

ι ◦ T n−1−iβ,α

where i ∈ {1, . . . , n}. Especially T nβ,α : In−1 → In−1 is topologically conjugate to

Tβn,α′ .

Proof. [66, p.2.21, Cor.] together with Lemma 8.3.1 and Lemma 8.2.7. Note that

In−1 ⊆ [zn−k, zn−k+1) and T = Tβ,α is bijective into the following directions

[zk, zk+1)T−1

← . . .T−1

← [z(−2k) mod n+1, z(−2k+1) mod n+1)T−1

← [zn−k, zn−k+1)T→ [zn, 1) ∪ [0, z1).

�

Figure 8.3: For (β, α) = (1.1190, 0.5409) ∈ D3,5 the Parry density is shown. It is

supported on five intervals (mod1). These depict the dynamical systems, which

are measure theoretically isomorphic to each other under iteration by T : x →{βx + α}.

8.4. SPECTRAL PROPERTIES 131

8.4 Spectral propertiesTwo notable works on β-transformations are Palmer, [66] and Glendinning, [33].Palmer investigated β-transformations with a focus on weak Bernoulliness, whileGlendinning did so for transitivity. Both will not play a role in what will be donein the following and therefore it will not be defined here. For a definition of weakBernoulli, see [66, Sec. 0, Def. 7] and for transitive the introduction of [33].Withthe techniques developed in Section 8.3 the spectral properties for intermediate β-transformations T on measure theoretical dynamical systems ([0, 1],B,T, µ) willbe investigated. Palmer has shown in [66] that T can only have eigenvalues whichare roots of unity and they cannot be larger than n, where β ≤ n√2, [66, page 2.9,2.13]. If we set nT ≔ lcm{λ : λ is an eigenvalue of T }, then T nT has only 1 as itseigenvalue, hence, with Lemma 6.5.3, it is the transformation of a weakly mixingdynamical system. By using that the tail-σ-algebras of T and T nT coincide and arefinite up to sets of µ-measure zero, Lemma 6.5.11, we know that the dynamicalsystem for T nT has finitely many ergodic components on which T nT restricted ontoa component is weakly mixing. Let us denote those components by W j, j ∈ J,then for the Perron-Frobenius operator of T nT one has

∫P( f ) dm|W j =

∫f dm|W j

for all f ∈ L1m([0, 1]) and P(h1W j) = h1W j , where j ∈ J, see also [40, Rem. 2].

This already nice result can be even more specified, but before that we need thefollowing

Theorem 8.4.1 ([66],[33],[40]). Let T (x) = {βx+α}, (β, α) ∈ ∆ be an intermediateβ-transformation.

(a) The measure theoretical dynamical system ([0, 1],B,T, µ) is weakly mixingif and only if supp(µ) = [0, 1], with countable many exceptional (β, α) ∈ ∆.

(b) The measure theoretical dynamical system ([0, 1],B,T, µ) is not weaklymixing if and only if it exists n, k ∈ N such that (β, α) ∈ Dk,n.

(c) The map T is not transitive if and only if (β, α) ∈ Dk,n for some k, n ∈ N.

(d) The measure theoretical dynamical system ([0, 1],B,T, µ) is not exact ifand only if there exists n, k ∈ N such that (β, α) ∈ Dk,n.

Proof. The first two points are already shown in [66], specifically Theorem 3 onpage 2.18 and the Corollary on page 2.21 and Theorem 4 on page 2.27. A more ap-proachable description of the sets Dk,n is given in [33, Prp. 2] as remarked before.While we are here interested in weak-mixing, Glendinning gives the statement interms of topological transitive, while Palmer did it for weak Bernoulli. But sheremarked on page 2.3, that in order to obtain her results she proves the system tobe weakly mixing, which is then followed to be weak Bernoulli, by an application


of Theorem B, which can be found on page 2.3. (Thm. B is a modified versionof [20, Thm. 1]). Indeed if one checks the proofs of Palmer, she proofs that thenumber of eigenvalues of T are exactly one eigenvalues or strictly larger that one,which is equivalent to weakly mixing, 6.5.3 and then concluded weak Bernoulliby implicitly using Theorem B. Another theorem she made use of is Theorem A,which holds in general and does not require weak Bernoulli and hence will notcause problems in our case.

(c) can be found in [33, Prp. 1,2] and for (d) notice that T admits inversebranches, the result is then due to (a),(b) and Proposition 6.5.12. �

A combination of Theorem 8.4.1 and Lemma 8.3.3 allow for a decompositionof return time combs acquired from β-transformations and will be done in the nexttheorem, after showing the following lemma.

Lemma 8.4.2. Denote by ϕi : Ii → [0, 1], i ∈ {1, . . . , n} the isomorphisms given inLemma 8.3.3, for some β-transformation T (x) = {βx + α}, where (β, α) ∈ Dk,n anddenote by µ the Parry measure to T . With these assumptions supp(µ) is a subsetof

⋃ni=1 Ii and µ(I j) = 1/n for all 1 ≤ j ≤ n.

Proof. By definition dk < zk < T (0) < dk+1 < T (1) < zk+1 < dk+2, where[dk, dk+1) = [wk]T and [dk+1, dk+2) = [wk+1]T for some finite words wk,wk+1 ∈

Ln(T ), by Lemma 8.1.1. By choice of the cylinder [w j]T , the mapping T n|[w j]T

isa bijection with fix point z j and (T n)′ = βn, j ∈ {k, k + 1}. This expanding be-haviour gives T n(x) < x for x < z j and x < T n(x) for z j < x, where j ∈ {k, k + 1}.Thus T (0) ≤ T n+1(0), T n+1(1) < T (1) and T n([T (0),T (1))) ⊆ [T (0),T (1)). Hencesupp(µ) = supp(hm) ⊆

⋃ni=1 Ii by definition of h, see (8.2). For the second part

notice that µ(Ii) = µ(I j) for all 1 ≤ i, j ≤ n, since every Ii ⊆ [zki , zki+1) for all1 ≤ i ≤ n − 1 and some 1 ≤ ki ≤ n − 1 and In ⊆ [zn, 1) ∪ [0, z1). Therefore setkn ≔ n and thus ki ≠ k j for all i ≠ j, 1 ≤ i, j ≤ n, the result then follows as µ is anT invariant probability measure and the intervals corresponding to zk are mappedinto each other by Lemma 8.2.7 for all 1 ≤ k ≤ n. �

Theorem 8.4.3. Let T (x) = {βx + α}, where (β, α) ∈ Dn,k and ([0, 1],B,T, µ) bea measure theoretical dynamical system with Parry measure µ = hm. Denote byηy an f1-weighted return time comb and by η′y an f2 weighted return time combwith respect to T and reference point y. Further let (Ii)n

i=1, ι,T = Tβ,α and α′ =ι(T n−1(0)) be as in Lemma 8.3.3, such that µ|Ii ≅ ν for all i ∈ {1, . . . , n}, whereν = hνm denotes the Parry measure to Tβn,α′ given by (8.2) on ([0, 1],B,Tβn,α′).Then for ν-a.e. y one has for any z ∈ Z, where 0 ≤ z = znn + r, 0 ≤ r < n, zn ∈ Z

ηy ~ ˜η′y(z) =1n

n∑i=1

νi,r ~ ν′i,0(zn).


Proof. Denote by ϕi : Ii → [0, 1] the isomorphisms given in Lemma 8.3.3, suchthat µ|Iiϕ

−1i = ν, where ν = hνm is the unique invariant probability measure on

([0, 1],B,Tβn,α′), α′ = ι(T n−1(0)), as in Lemma 8.3.3, and hν is given by (8.2) viaTβn,α′ . Then supp(µ) is contained in

⋃ni=1 Ii due to Lemma 8.4.2. Then for ν-a.e. y

one has for any z ≥ 0, where z = znn + r, 0 ≤ r < n, zn ∈ Z and 1 ≤ i ≤ n

ηy ~ ˜η′y(z) =∫ 1

0f1 ◦ T z · f2 dµ

=

n∑i=1

∫Ii

f1 ◦ T z · f2 dµ

=

n∑i=1

∫Ii

f1 ◦ T z ◦ ϕ−1i ◦ ϕi · f2 ◦ ϕ

−1i ◦ ϕi dµ

=

n∑i=1

µ(Ii)µ(Ii)−1∫ 1

0f1 ◦ T z ◦ ϕ−1

i · f2 ◦ ϕ−1i dµ|Ii ◦ ϕ

−1i

=1n

n∑i=1

∫ 1

0f1 ◦ T znn+r ◦ ϕ−1

i · f2 ◦ ϕ−1i dν

=1n

n∑i=1

∫ 1

0f1 ◦ T r ◦ ϕ−1

i ◦ T znβn,α′ · f2 ◦ ϕ

−1i dν.

(8.3)

Here we have used µ(Ii) = 1/n for all 1 ≤ i ≤ n by Lemma 8.4.2. In the same wayfor z < 0, where z = znn + r, 0 ≤ r < n, zn ∈ Z and 1 ≤ i ≤ n

ηy ~ ˜η′y(z) =∫ 1

0f1 · f2 ◦ T |z| dµ

=

∫ 1

0f1 · f2 ◦ T |zn |n−r dµ ◦ T−r

=

∫ 1

0f1 ◦ T r · f2 ◦ T |zn |n dµ

=

n∑i=1

µ(Ii)−1∫

Ii

f1 ◦ T r · f2 ◦ T |zn |n dµ

=

n∑i=1

µ(Ii)−1∫

Ii

f1 ◦ T r ◦ ϕ−1i ◦ ϕi · f2 ◦ T |zn |n ◦ ϕ−1

i ◦ ϕi dµ

=

n∑i=1

µ(Ii)µ(Ii)−1∫ 1

0f1 ◦ T r ◦ ϕ−1

i · f2 ◦ T |zn |n ◦ ϕ−1i dµ|Ii ◦ ϕ

−1i

=µ(In−1)n∑

i=1

∫ 1

0f1 ◦ T r ◦ ϕ−1

i · f2 ◦ ϕ−1i ◦ T |zn |

βn,α′ dν.


With that for all z ∈ Z, by definition of νi,r, ν′i,r for all 0 ≤ i − 1, r < n

νi,r ~ ν′i,r(z) =

⎧⎪⎪⎨⎪⎪⎩∫ 1

0f1 ◦ T r ◦ ϕ−1

i ◦ T zβn,α′ · f2 ◦ ϕ

−1i dν , z ≥ 0∫ 1

0f1 ◦ T r ◦ ϕ−1

i · f2 ◦ ϕ−1i ◦ T |z|βn,α′ dν , z < 0

.

�

For the remainder of this section we will use the following notations and defi-nitions.

Definition 8.4.4. For η ∈ C′c(Z), define

η(n,r)(z) ≔

⎧⎪⎪⎨⎪⎪⎩η(zn) , ∃zn ∈ Z : z = nzn + r0 , otherwise

With this association η(n,r) can be seen as a “stretching and shifting” of η, inparticular we have for any f ∈ Cc(Z)

⟨η(n,r), f ⟩ = ⟨η, f ◦ sn,r⟩,

where sb,a(x) ≔ bx + a for a, b ∈ Z and x ∈ Z or some subset of R. Also we setey(x) ≔ e2πixy, where x, y ∈ R or some subset of R.

Lemma 8.4.5. For η ∈ SFP(Z) the Bochner transform of η(n,r) is given by

(η(n,r))∧ = n−1 e−r

(ˆη ◦ s−11/n,0 ∗ δn−1Zn

)= n−1 (e−r(ˆη ◦ s−1

1/n,0)) ∗ (e−rδn−1Zn),

where s1/n,0 : [0, 1]→ [0, 1/n], x ↦→ x/n transports ˆη from [0, 1] to [0, 1/n].

In the following two proofs for Lemma 8.4.5 are presented, as the approachesin them are quite differently and a further comment is made in Remark 8.4.6.

Proof of Lemma 8.4.5. The main idea of the proof is to use Proposition 7.1.1,which enables us to apply the rule of integration by substitution. For that let η′

be a functional on Z and denote the embedding of the integers into the space offunctionals on the real numbers by η′R. With that ˆη′R = ˆη′ ∗ δZ by Definition 5.2.5and Proposition 7.1.1. For any f ∈ Cc(R) one has

⟨η(n,0)R , f ⟩ = ⟨ηR, f ◦ sn,0⟩ = ⟨ˆηR, ( f ◦ sn,0)∨⟩.

An application of integration by substitution yields

⟨ˆηR, ( f ◦ sn,0)∨⟩ =∫R

( f ◦ sn,0)∨(x) dˆηR(x)


=

∫R

∫R

f (ny) e2πiyx dm(y) dˆηR(x)

=

∫R

1n

∫R

f (y) e2πi(y)n−1 x dm(y) dˆηR(x)

=1n

∫R

f ∨(x/n) dˆηR(x)

=1n

∫R

f ∨ ◦ s1/n,0(x) dˆηR(x)

=1n

∫R

f ∨(x) dˆηR ◦ s−11/n,0(x),

by definition of η. Note that for any f ∈ Cc(R)

⟨n−1 ˆηR ◦ s−11/n,0, f ⟩

=⟨n−1ˆη ∗ δZ ◦ s−11/n,0, f ⟩

=1n

∫R

f ◦ s1/n,0(x) d(ˆη ∗ δZ)(x)

=1n

∫Z

∫[0,1]

f(x + y

n

)dˆη(x) dδZ(y)

=1n

∫Z

∫Zn

∫[0,1]

f(x + nz + t

n

)dˆη(x) dδZn(t) dδZ(y)

=1n

∫Z

∫n−1Zn

∫[0,1]

f(s1/n,0(x) + t + z

)dˆη(x) dδn−1Zn(t) dδZ(y)

=1n

∫Z

∫n−1Zn

∫[0,1/n]

f (x + t + z) dˆη ◦ s−11/n,0(x) dδn−1Zn(t) dδZ(y)

=

∫Z

∫[0,1]

f (y + z) dn−1(ˆη ◦ s−1

1/n,0

)∗ δn−1Zn(y) dδZ(y)

=⟨(

n−1ˆη ◦ s−11/n,0 ∗ δn−1Zn

)∗ δZ, f

⟩,

which by another application of Proposition 7.1.1, proofs the claim for r = 0.Note that in this case η(n,0) ∈ SFP(Z), which is due to ⟨η(n,0)

R , f ⟩ = ⟨ηR, f ◦ sn,0⟩

and the function f ◦ sn,0 is positive definite if f is positive definite as a directconsequence of Definition 5.2.1. For r ≠ 0 the claim follows by an application thefirst chain of equalities in the proof of Lemma C.3.2 by

⟨η(n,r), f ⟩ = ⟨η, f ◦ s1,r ◦ sn,0⟩

= ⟨ˆη, ( f ◦ s1,r ◦ sn,0)∨⟩

= ⟨n−1ˆη ◦ s−11/n,0 ∗ δn−1Zn , ( f ◦ s1,r)∨⟩

= ⟨e−r (n−1ˆη ◦ s−11/n,0 ∗ δn−1Zn), f ∨⟩.


The second equality in the statement holds as e−r(x + t) = e−r(x)e−r(t). �

Proof of Lemma 8.4.5. By making use of Theorem 5.2.7(3), the Bochner trans-form of η(q) is uniquely determined by the following equation, where one has forany f = f1 ∗ ˜f1 and f1 ∈ Cc([0, 1))

⟨η(n,0), ˆf ⟩ = ⟨η(n,0), ˆf ◦ sn,0⟩.

Here ˆf ◦ sn,0 is a function on Z and hence for any z ∈ Z given by

ˆf ◦ sn,0(z) =∫ 1

0f (y)e−2πinzy dm(y)

=

n−1∑l=0

∫ (l+1)/q

l/nf (y)e−2πinzy dm(y)

=

n−1∑l=0

∫ 1/n

0f (y + l/q)e−2πinz(y+l/n) dm(y)

=

n−1∑l=0

1n

∫ 1

0f (w/q + l/q)e−2πizw dm(w)

=

n−1∑l=0

1n

∫ 1

0gl(w)e−2πizw dm(w) =

n−1∑l=0

1nˆgl(z),

where gl(w) ≔ f (w/q+ l/q). With that we have by linearity and Theorem 5.2.7(3)

⟨η(n,0), ˆf ⟩ = n−1∑l=0

1n⟨η,ˆgl⟩ =

n−1∑l=0

1n⟨ˆη, gl⟩ =

n−1∑l=0

1n

∫ 1

0gl(w) dˆη(w)

=

n−1∑l=0

1n

∫ 1

0f (w/n + l/n) dˆη(w) =

⟨1qˆη ◦ s−1

1/n,0 ∗ δq−1Zq , f⟩,

which concludes the proof for r = 0. In the case r ≠ 0 we refer to the formerproof. �

Remark 8.4.6. Note that the argument w/n + l/n in the function f in the proofsof Lemma 8.4.5, which leads to ˆη ◦ s−1

1/n,0 ∗ δq−1Zq is considered on the unit intervalmodulo one and hence can also be interpreted on the unit circle. In this sense{w/n + l/n : 0 ≤ l < n} is then identified with the n-th roots w. Thus insteadof writing ˆη ◦ s−1

1/n,0 ∗ δn−1Zn we can also write ˆη ◦ (ϕn)−1, where ϕ : w ↦→ wn forw ∈ {x ∈ C : ∥x∥ = 1} and ϕ−1(w) = n

√w.


Definition 8.4.7. Let m ∈ N and choose natural numbers ki < ni with gcd(ki, ni) =1, thus Dki,ni is well-defined for all 0 ≤ i ≤ m. Set ℓ ≔

((ki, ni)

)mi=0 and define

Dℓ ≔ {(β0, α0) : (β, α) ∈ ∆} ,

where (β, α) ↦→ (β′, α′) given by Lemma 8.3.1 and (βm, αm) ≔ (β′, α′), (βi, αi) ≔((βi+1)′, (αi+1)′

)for all 1 ≤ i ≤ m − 1.

Remark 8.4.8. It is notable that the map (β, α) ↦→ (β′, α′) given by Lemma 8.3.1is a bijection. For a fixed sequence

((βi, αi)

)mi=0 the mapping (βi, αi) ↦→ (βi+1, αi+1)

is done via Lemma 8.3.3, while (βi+1, αi+1) ↦→ (βi, αi) is done via Lemma 8.3.1 for1 ≤ i ≤ m − 1. Further, β0 ≤

m√2 and Dℓ ⊆ Dk0,n0 . In Figure 8.4 some Dℓ areshown with their true proportions to each other, while in Figure 1.1 examples ofDℓ are shown for some ℓ to emphasise the picture one should keep in mind.

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

0.25 0.3 0.35 0.4 0.45 0.5

Figure 8.4: An plot of Dℓ, whereas ℓ is chosen to be ((1, 2)) and ((1, 2); (1, 3)).

Example 8.4.9. Consider ℓ = ((1, 2), (1, 3), (3, 5)) and fixed (β, α), (βi, αi) for i ∈{0, 1, 2}, in accordance to Definition 8.4.7 by the mapping (β, α) ↦→ (β′, α′) definedin Lemma 8.3.1. Then (β0, α0) ∈ Dℓ, (β1, α1) ∈ D((1,3),(3,5)) and (β2, α2) ∈ D3,5.One can then define the tuple (β3, α3) ∈ ∆ by applying Lemma 8.3.3 and this tuplemight not be in any Dk,n for all k, n ∈ N. Further note that D((1,3),(3,5))∩Dℓ = ∅, sinceD((1,3),(3,5)) ⊆ D1,3 and Dℓ ⊆ D1,2, which have empty intersection by definition.

With the new notions Lemma 8.4.5 can be strengthened to the following theo-rem.


Theorem 8.4.10. Let ℓ ≔((ki, ni)

)mi=0 with gcd(ki, ni) = 1, (β0, α0) ∈ Dℓ and define

(βi, αi) according to Definition 8.4.7 via Lemma 8.3.3 and set Ti(x) ≔ {βix + αi}

with Parry measure µi for all 0 ≤ i ≤ m + 1. The choice of (βi, αi) induces afinite sequence of measure theoretical isomorphisms (ϕi, j)

nij=1 : Ii, j → [0, 1] given

by ιi ◦ T ni−1− ji for each 0 ≤ i ≤ m and (µi)|Ii, j ◦ ϕi, j ≅ 1/niµi+1 for all 0 ≤ i ≤ m,

1 ≤ j ≤ ni.Denote by ηy an f1-weighted return time comb and by η′y an f2-weighted return

time comb, both with respect to T0 and reference point y. Then for µ0-a.e. y onehas for any z ∈ Z

ηy ~ ˜η′y(z) =1q

∑i∈

✕mj=0{1,...,n j}

q−1∑r=0

(νi,r ~ ˜ν′i)(q,r)

(z),

where q ≔∏m

j=0 n j. Here νi,r is an f1 ◦ T r0 ◦ φ

−1i -weighted return time comb with

respect to Tm+1 and ν′i is an f2 ◦ φ−1i -weighted return time comb with respect to

Tm+1, where i ∈✕m

j=0{1, . . . , n j} and 0 ≤ r ≤ (∏m

j=0 n j) − 1 such that for all z ∈ Z

νi,r ~˜(ν′i)(z) =

⎧⎪⎪⎨⎪⎪⎩∫ 1

0f1 ◦ T r

0 ◦ φ−1i ◦ T z

m+1 · f2 ◦ φ−1i dµm+1 , z ≥ 0∫ 1

0f1 ◦ T r

0 ◦ φ−1i · f2 ◦ φ

−1i ◦ T |z|m+1 dµm+1 , z < 0

,

where φi ≔ ϕm,im ◦ . . . ◦ ϕ0,i0 .

Proof. The proof is an inductive application of (8.3). Note that for ν-a.e. y andany z ≥ 0

ηy ~ ˜η′y(z) =∫ 1

0f1 ◦ T z

0 · f2 dµ0 (8.4)

=1n0

n0∑i=1

∫ 1

0f1 ◦ T z

0 ◦ ϕ−10,i · f2 ◦ ϕ

−10,i dµ1

=1

n0n1

n0∑i1=1

n1∑i0=1

∫ 1

0f1 ◦ T z

0 ◦ ϕ−10,i0 ◦ ϕ

−11,i1 · f2 ◦ ϕ

−10,i0 ◦ ϕ

−11,i1 dµ2

=

m∏j=0

1n j

∑i∈

✕mj=0{1,...,n j}

∫ 1

0f1 ◦ T z

0 ◦ φ−1i · f2 ◦ φ

−1i dµm+1,

where φi = ϕm,im ◦ . . . ◦ ϕ0,i0 . In the following z ∈ Z will be decomposed, bychoosing a z0 ∈ Z, 0 ≤ r0 < n0 such that z = z0n0 + r0. Then choose recursivelyzi ∈ Z, 0 ≤ ri < ni such that zi−1 = zini + ri for all 1 ≤ i ≤ m. With thatz = zm

∏mj=0 n j + r0 +

∑mj=1 r j

∏ j−1i=0 ni and if we set Rz ≔ r0 +

∑mj=1 r j

∏ j−1i=0 ni, then

ηy ~ ˜η′y(z) =m∏

j=0

1n j

∑i∈

✕mj=0{1,...,n j}

∫ 1

0f1 ◦ T

zm∏m

j=0 n j+Rz

0 ◦ φ−1i · f2 ◦ φ

−1i dµm+1


=

m∏j=0

1n j

∑i∈

✕mj=0{1,...,n j}

∫ 1

0f1 ◦ T Rz

0 ◦ φ−1i ◦ T zm

m · f2 ◦ φ−1i dµm+1, (8.5)

for any z ≥ 0. In the case z < 0 we have

ηy ~ ˜η′y(z) =∫ 1

0f1 · f2 ◦ T |z|0 dµ0

=

∫ 1

0f1 · f2 ◦ T

|zm |∏m

j=0 n j−Rz

0 dµ0

=

∫ 1

0f1 ◦ T Rz · f2 ◦ T

|zm |∏m

j=0 n j

0 dµ0

=

m∏j=0

1n j

∑i∈

✕mj=0{1,...,n j}

∫ 1

0f1 ◦ T Rz ◦ φ−1

i · f2 ◦ T|zm |

∏mj=0 n j

0 ◦ φ−1i dµm+1

=

m∏j=0

1n j

∑i∈

✕mj=0{1,...,n j}

∫ 1

0f1 ◦ T Rz ◦ φ−1

i · f2 ◦ φ−1i ◦ T |zm |

m dµm+1

Every one of them is an averaged convolution of an f1 ◦ T r0 ◦ φ

−1i -weighted re-

turn time comb νi,r and an f2 ◦ φ−1i -weighted return time comb ν′i , where i ∈✕m

j=0{1, . . . , n j} and 0 ≤ r ≤ (∏m

j=0 n j) − 1 such that for all z ∈ Z

νi,r ~˜(ν′i)(z) =

⎧⎪⎪⎨⎪⎪⎩∫ 1

0f1 ◦ T r

0 ◦ φ−1i ◦ T z

m+1 · f2 ◦ φ−1i dµm+1 , z ≥ 0∫ 1

0f1 ◦ T r

0 ◦ φ−1i · f2 ◦ φ

−1i ◦ T |z|m+1 dµm+1 , z < 0

�

The functionals described via Theorem 8.4.10 can be transformed by an ap-plication of Theorem 6.5.13, which yields the following theorem.

Theorem 8.4.11. Let ηy ~ ˜η′y be the functional given in Theorem 8.4.10 whereadditionally f1, f2 ∈ BV are real-valued and assume (βm+1, αm+1) ∉ Dk,n for allk, n ∈ N, gcd(k, n) = 1. Then its Bochner transform is given by

(ηy ~ ˜η′y)∧ = 1q2

∑i∈

✕mj=0{1,...,n j}

q−1∑r=0

Ci,re−r δ 1qZq+ qe−r(gi,r ◦ sq,0m|[0,1/q)) ∗ δ 1

qZq,

where sb,a : x ↦→ bx + a for a, b ∈ R, Ci,r ∈ R is a constant and gi,r an analyticfunction for all 0 ≤ r ≤ q − 1 and i ∈

✕mj=0{1, . . . , n j}. In particular Ci,r ≔∫

f1 ◦ T r0 ◦ φ

−1i dµm+1

∫f2 ◦ φ

−1i dµm+1 and gi,r(x) ≔

∑z∈Z cz e2πixz, with cz ≔∫

f1◦T r0◦φ

−1i ·Ψ

|z|( f2◦φi ·h) dm, for z > 0, cz ≔∫

f2◦φ−1i ·Ψ

|z|( f1◦T r0◦φ

−1i ·h) dm, for


z < 0 and c0 ≔∫

f1◦T r0 ◦φ

−1i · f2◦φ

−1i dµm+1−

∫f1◦T r

0 ◦φ−1i dµm+1

∫f2◦φ

−1i dµm+1

for all 0 ≤ r ≤ q−1 and i ∈✕m

j=0{1, . . . , n j}. Here h andΨ are chosen with respectto ([0, 1),B,Tm+1, µm+1).

Proof. For all 0 ≤ r ≤ q − 1, the Bochner transform of (νi,r ~ ˜ν′i)(q,r) is given by((νi,r ~ ˜ν′i)(q,r)

)∧= q−1

(e−r(νi,r ~ ˜ν′i)∧ ◦ s−1

1/q,0

)∗ e−rδ 1

qZq, (8.6)

where Lemma 8.4.5 were used. Since (βm+1, αm+1) ∉ Dk,n for all k, n ∈ N, the dy-namical system ([0, 1),B,Tm+1, µm+1) is weakly mixing. An application of Theo-rem 6.5.13 then yields

(νi,r ~ ˜ν′i)∧ = Ci,r δ0 + gi,rm|[0,1],

where Ci,r ≔∫

f1 ◦ T r0 ◦ φi dµm+1

∫f2 ◦ φi dµm+1 and gi,r(x) ≔

∑z∈Z cz e2πixz, with

cz ≔∫

f1 ◦ T r0 ◦ φ

−1i · Ψ

|z|( f2 ◦ φ−1i · h) dm, for z > 0, cz ≔

∫f2 ◦ φ

−1i · Ψ

|z|( f1 ◦

T r0 ◦ φ

−1i · h) dm, for z < 0 and c0 ≔

∫f1 ◦ T r

0 ◦ φ−1i · f2 ◦ φ

−1i dµm+1 −

∫f1 ◦ T r

0 ◦

φ−1i dµm+1

∫f2 ◦ φ

−1i dµm+1 is an analytic function. As for all f ∈ Cc([0, 1])⟨

e−r(νi,r ~ ˜ν′i)∧ ◦ s−11/q,0, f

⟩=

⟨e−r

(Ci,r δ0 + gi,rm|[0,1]

)◦ s−1

1/q,0, f⟩

=

∫ 1

0f (s1/q,0(x)) e−r(s1/q,0(x)) d(Ci,r δ0 + gi,rm|[0,1]) (x)

= Ci,r ⟨δ0, (e−r f ) ◦ s1/q,0⟩ +

∫ 1

0f (x/q) e−r(x/q) gi,r(x) dm|[0,1])(x)

= Ci,r ⟨δ0, f ⟩ + q∫ 1

0f (x) e−r(x) gi,r(qx) dm|[0,1/q))(x)

we have for (8.6)((νi,r ~ ˜ν′i)(q,r)

)∧= q−1

(Ci,r δ0 + q e−rgi,r ◦ sq,0m|[0,1/q)

)∗ (e−rδ 1

qZq)

= q−1(Ci,r δ 1

qZq+ q(e−rgi,r ◦ sq,0m|[0,1/q)) ∗ (e−rδ 1

qZq))

= q−1(Ci,re−r δ 1

qZq+ qe−r(gi,r ◦ sq,0m|[0,1/q)) ∗ δ 1

qZq

)With that it follows

(ηy ~ ˜η′y)∧ = 1q

∑i∈

✕mj=0{1,...,n j}

q−1∑r=0

((νi,r ~ ˜ν′i)(q,r)

)∧


=1

q

∑i∈�m

j=0{1,...,n j}

q−1∑r=0

q−1(e−r(νi,r � ν′i)

∧ ◦ s−11/q,0

)∗ (e−rδ 1

qZq)

=1

q2

∑i∈�m

j=0{1,...,n j}

q−1∑r=0

Ci,re−r δ 1qZq+ qe−r(gi,r ◦ sq,0m|[0,1/q)) ∗ δ 1

qZq.

�

Figure 8.5: The left side shown the sets D1,3 and D((1,3);(1,2)). The marked points

are (α0, β0) = (0.2324, 1.2079) and (α1, β1) = (0.2839, 1.1001). In Figure 8.6 a

spectral return measure is given with respect to these points. The same is shown

in Figure 8.7 for the right side, where (α0, β0) = (0.1434, 1.1190) respectively

(α1, β1) = (0.1799, 1.0385) are points inside D1,5 respectively D((1,5);(1,3)).

Figure 8.6: The spectral return measures for (α0, β0) = (0.2324, 1.2079) and

(α1, β1) = (0.2839, 1.1001) with fi = 1[ci,1) chosen respectively, where ci =

(1 − αi)/βi for i ∈ {0, 1}. It illustrates the embedding of D1,2 into D1,3. The la-

belling on the x-axis marks the discrete part of the spectral return measures.


Figure 8.7: The spectral return measures for (α0, β0) = (0.1434, 1.1190) and

(α1, β1) = (0.1799, 1.0385) with fi = 1[ci,1) chosen respectively, where ci =

(1 − αi)/βi for i ∈ {0, 1}. It illustrates the embedding of D1,3 into D1,5. The

right figure is also shown in Figure 1.3. The continuous part here is hardly visible

compared to the one in Figure 8.6.

Remark 8.4.12. If (β0, α0) ∈ Δ there exists always an � such that (β0, α0) ∈ D� and

(βm+1, αm+1) � Dk,n for all k, n ∈ N, gcd(k, n) = 1, where βm+1 = βq0, q =

∏mj=0 nj.

This is as any β-transformation with slope greater than√

2 is weakly mixing, due

to [20, Thm. 1, Thm. 2], or as√

2 = argmaxβ{(β, α) ∈ Dn,k : gcd(n, k) = 1, (β, α) ∈Δ}. In fact, if one sets Δ′ � Δ\

⋃k<n,gcd(k,n)=1 Dk,n and defines

D′� � {(β0, α0) : (β, α) ∈ Δ′} ,

for any sequence � �((ki, ni)

)mi=0 for some m ∈ N, where ki < ni with gcd(ki, ni) =

1 for all 0 ≤ i ≤ m. Then Δ =⋃� D′� and for any (β0, α0) ∈ D′

� ⊆ D� the

induced tuple (βm+1, αm+1) describes a weak mixing β-transformation, so that The-

orem 8.4.11 can be applied.

Corollary 8.4.13. The spectral return measure of any β-transformation is the sumof spectral return measures of a weakly mixing dynamical system for a certain β-transformation, which admits an explicit description in terms of Theorem 8.4.10.In particular the spectral return measure of any β-transformation decomposesinto a Lebesgue absolutely continuous part and a finite discrete part.

8.5 Substitutions from β-transformationsIn this section we connect β-transformations to rotation maps via Lemma 8.3.3,

which are then related to substitutions that were defined in Section 2.3. Finally,

the spectral (return) measures of these substitutions is discussed.

8.5. SUBSTITUTIONS FROM β-TRANSFORMATIONS 143

Lemma 8.5.1. Let k, n ∈ N+ with 1 ≤ k < n and gcd(k, n) = 1. Let Tk/n(x) ={k/n + x} be the rational rotation by k/n on [0, 1) and let (β, α) ∈ Ck,n. For anyx ∈ [γ, zn−k+1) the following assertion

1[γ,1)(T j(x)) = 1[1− kn ,1)

(T j

k/n(1 − kn )

)= 1{0,...,k−1}( jk mod n) (8.7)

holds for each j ∈ {0, . . . , n − 1} and for x ∈ [zn−k, γ)

1[γ,1)(T j(x)) = 1(1− kn ,1]

(T j

k/n(1 − kn )

)= 1{1,...,k}( jk mod n) (8.8)

for each j ∈ {0, . . . , n − 1}.

Proof. The statement is a consequence of Lemma 8.2.6.iii) and Lemma 8.2.7. �

The map is exactly the same as in Definition 2.3.7, thus by Theorem 2.3.8 thesequence coincides with ωn

l of Definition 2.3.2, where l ∈ {0, 1}. This motivatesthe following definition and ensures it is well-defined.

Definition 8.5.2. Define

Q ≔{τa1ρa2τa3 . . . τa j−1ρa j−1τTM : ∀ j ∈ 2N+ and (ai)

ji=1 ∈ N × N

j−1+

}∪{

τa1ρa2τa3 . . . ρa j−1τa j−1τTM : ∀ j ∈ (2N + 1) and (ai)ji=1 ∈ N × N

j−1+

}.

For any Dk,n there is a unique substitution σ ∈ Q such that κk/nl = ωn

l = σ(l),where l ∈ {0, 1}, which is called the substitution associated with Dk,n. We mayalso say that Dk,n is associated with σ.

Lemma 8.5.3. • Any σ ∈ Q is a primitive substitution of constant length.

• (σ(l))0 = l, where l ∈ {0, 1}.

• σn(l) is a prefix of σn+1(l) for any n ∈ N, l ∈ {0, 1}.

Proof. Example 2.3.4 implies |σ(0)| = |10S 2σ(1)| and |σ(1)| = |01S 2σ(0)|, whichalso shows that σ is primitive. The other two claims also are also covered inSection 2.3. �

Lemma 8.5.4. Let ℓ ≔((ki, ni)

)mi=0 with gcd(ki, ni) = 1, let Dki,ni be associated with

σi for all 0 ≤ i ≤ m and for (β, α) ∈ Dℓ let T (x) = {βx + α} be a β-transformationwith T-discontinuity γ. There exists non-empty open intervals I, J such that

1[γ,1)(T j(x)) =(σ0 . . . , σm(1)

)j,

for all x ∈ I ∪ {γ} and

1[γ,1)(T j(x)) =(σ0 . . . , σm(0)

)j,

for all x ∈ J, where j ∈ {0, . . . , (∏m

i=0 ni) − 1}.


Proof. That the statement holds for γ is by construction, as presented in Lem-mas 8.3.1 and 8.3.3. Further x ∈ I ⊆ [zn0−k0 , zn0−k0+1) due to zk ≤ T (0) < T (1) ≤zk0+1. Let ℓ j ≔

((ki, ni)

) ji=0 for all 0 ≤ j ≤ m, thus ℓm = ℓ. By definition,

(β, α) ∈ Dk0,n0 and hence by Lemma 8.5.1 and Theorem 2.3.8

1[γ,1)(T j(x)) = 1{0,...,k0−1}( jk0 mod n0) =(σ0(1)

)j, j ∈ {0, . . . , n0 − 1}.

By Lemma 8.3.3 T is topologically conjugate to T1(x) = {β1x + α1} and (β1, α1) ∈Dℓm−1 ⊆ Dk1,n1 ⊆ ∆. In the same way as before 1[γ1,1)(T

j1(x)) =

(σ1(1)

)j for

j ∈ {0, . . . , n1 − 1} and x ∈ [γ1, zn1−k1+1), where γ1 denotes the T1-discontinuity.With that (T n0 s)|In0−1(x) = T s

1(ι1x) for any s ∈ N, where In0−1, ι1 are chosen as inLemma 8.3.1. Thus either (8.7) or (8.8) is chosen for the next n0 iterates for Tdepending on T s

1(x) = 1{0,...,k1−1}(sk1 mod n1) attaining the value 0 or 1, by (8.7).Then for any x ∈ ι−1

1 ([γ1, zn1−k1+1)) one has

1[γ,1)(T i(x)

)=

(σ0σ1(1)

)i, i ∈ {0, . . . , n0n1 − 1}.

An inductive application yields the result for the case x ∈ ι1◦· · ·◦ιm([γm, znm−km+1))and an analogue discussion for x ∈ ι1 ◦ · · ·◦ ιm([znm−km , γ)) proofs the second claim.Especially both intervals are non-empty and γ ∈ ι1 ◦ · · · ◦ ιm([γm, znm−km+1)), asthe discontinuity and hence the T -discontinuity is preserved under topologicalconjugacy. �

Corollary 8.5.5. Let ℓ ≔((ki, ni)

)mi=0 with gcd(ki, ni) = 1, let Dki,ni be associated

with σi for all 0 ≤ i ≤ m. For (β, α) ∈ Dℓ let T (x) = {βx+α} be a β-transformationand denote by µ the Parry measure to T . Let l ∈ {0, 1}, then for µ-almost everyx ∈ [0, 1) there exists an s ∈ N such that

1[γ,1)(T s+ j(x)) =(σ0 . . . , σm(v)

)j,

for some v ∈ {0, 1}N.

Proof. Let µm be the Parry measure to Tm(x) = {βmx + αm}, where (βm, αm) ∈Dkm,nm . By Lemma 8.4.2 the support of µm is contained in the disjoint union⋃nm

i=1 Ii of the intervals given by Lemma 8.3.3. Especially Ii is measure theo-retically isomorphic to I j for all i, j ∈ {1, . . . , nm} and hence each Ii containsa set of non-zero measure. By construction ι1 ◦ · · · ◦ ιm(Inm−1) is a subset ofι1 ◦ · · ·◦ ιm([znm−km , znm−km+1)) from the proof of Lemma 8.5.4 and µm(Inm−1) ≠ 0 onealso has µ(ι1 ◦ · · · ◦ ιm(Inm−1)) ≠ 0 due to them being measure theoretically isomor-phic up to a constant. The statement then follows by ergodicity of µ, as then afterfinitely many steps, e.g. s ∈ N, µ-almost surely T s(x) ∈ ι1 ◦ · · · ◦ ιm(Inm−1). Thenone of the formulas given in Lemma 8.5.4 holds and this step is repeated everytime, as the dynamical system to T

∏mi=0 ni restricted on ι1 ◦ · · · ◦ ιm(Inm−1) is measure

theoretically isomorphic to the dynamical system of Tm. Especially the sequencev in the statement of the theorem is generated by v j = 1[γm,1) ◦T j

m for all j ∈ N. �


Remark 8.5.6. In prospect of Corollary 8.5.5 and its proof, by definition, the T -discontinuity is always an element of ι1 ◦ · · · ◦ ιm(Inm−1) and mapped to γm withthe conjugacy maps. If one sets v j = 1[γm,1) ◦ T j

m(γm) for all j ∈ N, then for s = 0the formula given in Corollary 8.5.5 is exactly the kneading sequence of T . Forfurther information see [33].

The next aim is to link the autocorrelation of a β-transformation T (x) = {βx +α}, where (β, α) ∈ Dℓ to the autocorrelation given by substitutions, which will bespecified below.

Lemma 8.5.7. For any (σi)i∈N ∈ QN the subshift Xul is minimal, where ul =

limm→∞ σ0 . . . σm(l) for l ∈ {0, 1}. Especially ul always exists and Xu0 = Xu1 .

Proof. To see that ul always exists is due to an induction by using that σ(a)0 = afor all σ ∈ Q and a ∈ {0, 1}. As a prefix of σ0σ1(l) is given by σ0(l) and then bythe inductive step a prefix of σ0 . . . σm+1(l) is σ0 . . . σm(l), which converges withrespect to the ultrametric of {0, 1}N. With that one also has (ul)0 = l. Let w be afactor of ul, then it exists an m ∈ N such that w is a factor of σ0 . . . σm(l) as l isa letter of σm+1(0) and σm+1(1) one has that w is a factor of σ0 . . . σm+1 appliedon either 0 or 1 and thus w occurs infinitely often with bounded gaps in ul. Withthat the subshifts Xu0 and Xu1 are minimal and they are equal as every factor ofσ0 . . . σm(l) is a factor of σ0 . . . σm

((θl)l

), where (θl)l = (uθl)[0,1]. �

There exists a more general version of the previous lemma in [14, Thm. 5.2],that also presents an “if and only if” relation for so called S-adic representations,where a u ∈ ΣN admits an S-adic representation if there exists a sequence of sub-stitutions (σi)i∈N on Σ such that u = limm→∞ σ0 . . . σm(a) for some a ∈ Σ. Further,if for all a ∈ Σ the limit | limm→∞ σ0 . . . σm(a)| = ∞ and for all m ∈ N exists ann ∈ N such that a occurs in σm . . . σm+n(b), for any b ∈ Σ the sequence (σi)i∈N iscalled everywhere growing and weakly primitive. Note that these properties arefulfilled for all sequences in QN (They were already used several times, e.g. inthe proof of Lemma 8.5.7).

For a sequence (σi)i∈N ∈ QN, let ni be the constant length of σi and ki =

|σi(0)|1 = σi(1)|1 for all i ∈ N. With that ni − ki = |σi(0)|0 = |σi(1)|0 for all i ∈ Nand one has for the incidence matrix of σi

M(σi) =⎛⎜⎜⎜⎜⎝σki,ni(0)

0

σki,ni(1)

0

σki,ni(0)1

σki,ni(1)

1

⎞⎟⎟⎟⎟⎠ = (ni − ki ni − ki

ki ki

)and thus it is of rank one. This also holds for the product of incidence matrices, as

M(σi)M(σ j) =((ni − ki)(n j − k j + k j) (ni − ki)(n j − k j + k j)

ki(n j − k j + k j) ki(n j − k j + k j)

)= n jM(σi).


for all i, j ∈ N. Hence for all j ∈ N the space

⋂m∈N

⎛⎜⎜⎜⎜⎜⎜⎝ j+m∏i=n

M(σi)

⎞⎟⎟⎟⎟⎟⎟⎠R2+ = M(σ j)R2

+,

is one-dimensional, which let us apply [14, Theorem 5.7] to yield that Xu isuniquely ergodic, where u = limm→∞ σ0 . . . σm(l), and l ∈ {0, 1}, see also [83].

Theorem 8.5.8. Let ℓ ≔((ki, ni)

)i∈N with gcd(ki, ni) = 1, let Dki,ni be associated

with σi for all i ∈ N. Set ℓm ≔((ki, ni)



of the 1[γm,1)-weighted return time combs with respect to the Parry measure µm andreference point ym, one has

v-limm→∞

γTm = γu,

where u = limm→∞ σ0 . . . σm(1), the subshift Xu is uniquely ergodic and γu denotesthe autocorrelation of u.

Proof. First we give a short repetition of the connection between autocorrelationand frequencies. Let X = {0, 1}N, cylinder sets taken in the following are con-sidered with respect to X. Further let v(m)

i = 1[γm,1) ◦ T im(ym) for all i ∈ N and

v(m) = (v(m)i )i∈N. Assume without loss of generality that z ≥ 0, as the the case z < 0

follows analogously. For n ∈ N the following chain of equalities holds.∑0≤n≤N

(1[γm,1) · 1[γm,1) ◦ T z

m

)(T n

m(ym)) =∑

0≤n≤N

(1[1] · 1[1] ◦ S z) (S n(v(m)))

=∑

0≤n≤N

(1[1] · 1

⋃w∈L|z|(X)[w1]

)(S n(v(m)))

=∑

0≤n≤N

1⋃w∈Lz−1(X)[1w1](S n(v(m)))

=∑

w∈Lz−1(X)

v(m)|n

1w1

.

The left hand side is a part of (6.2) and thus

γTm(z) = limN→∞

1N + 1

∑0≤n≤N

(1[γm,1) · 1[γm,1) ◦ T z

m

)(T n

m(y))

= limN→∞

1N + 1

∑w∈L|z|−1(X)

v(m)|N+1

1w1

=∑

w∈L|z|−1(X)

f1w1(v(m)),

(8.9)


for all z ∈ Z\{0} and for z = 0 one has γTm(0) = f1(ν(m)). This is, as for µm-almostevery x ∈ [0, 1], by Birkhoff’s ergodic theorem∫

1[w]Tmdµm = lim

N→∞

1N

N−1∑n=0

1[w]Tm◦ T n

m(x) = limN→∞

1N

N−1∑n=0

1[w] ◦ S nm(v) = fw(v),

where v = (1[w]Tm◦ T n

m(x))n∈N. By choice of Tm the sequence v(m) is given in termsof Corollary 8.5.5, that is S sv(m) = σ0 . . . σm(vm) for some vm ∈ X, s ∈ N andwith that fw(v(m)) = fw(σ0 . . . σm(vm)) for any w ∈ L|z|+1(X). Now we proof thetheorem. Set sm ≔

∏mi=0 ni and note that σi is a primitive substitution of length ni

for all i ∈ N. Especially |σ(0)|1 = |σ(1)|1 and is equal to km if km/nm ≤ 1/2 andnm − km otherwise. Take note that for any l ∈ {0, 1}

σ0 . . . σm−1(σmvm)|smN

w

smN

S Nkm |σ0 . . . σm−1(1 − l)|w + (nm − km) |σ0 . . . σm−1(l)|w

smN± N

2|w| − 2smN

=km |σ0 . . . σm−1(1 − l)|w + (nm − km) |σ0 . . . σm−1(l)|w

nmsm−1±

2|w| − 2sm

(8.10)

where N ∈ N and for N → ∞ one has an upper and a lower bound for anyaccumulation point of the sequence. Since u = limm→∞ σ0 . . . σm−1(1 − l) =limm→∞ σ0 . . . σm−1(l) and Xu is minimal and uniquely ergodic, the frequency fw(u)exists for all w ∈ L(X), but may, of course, only be non-zero for w ∈ L(Xu). Letfor z ∈ Z\{0}, w ∈ L|z|−1(Xu) and ε > 0, then it exists M1,M2 ∈ N such that

1sm−1|σ0 . . . σm−1(l)|w − fw(u)

< ε 2−|z|

for all l ∈ {0, 1}, where m ≥ M1 and (2|w| − 2)/sM2 ≤ ε 2−|z|, since sm → ∞ form→ ∞. Set M ≔ max{M1,M2}, then it follows with (8.10) for all m ≥ M

γTm(z) S∑

w∈L|z|−1(Xu)

f1w1(u) ± 2ε 2−|z| S γu(z) ± ε.

The right hand side is exactly the definition of γu and as pointwise convergenceis equivalent to vague convergence on the integers, the result follows for all z ≠ 0and the case z = 0 follows in a similar manner. �

Proposition 8.5.9. For sequences((km, nm)

)m∈N with gcd(km, nm) = 1, let Dkm,nm

be associated with σm for all m ∈ N. Given ym ∈ [0, 1) and (βm, αm) ∈ Dkm,nm ,m ∈ N we define the map Tm(x) ≔ {βmx + αm} with Tm-discontinuity γm. Iflimm→∞ km/nm = α and nm → ∞ for m → ∞, the autocorrelations γTm of the


1[γm,1)-weighted return time combs with respect to the Parry measure µm and ref-erence point ym converge vaguely to

v-limm→∞

γTm = γTα ,

where γTα is the autocorrelation of the rotation Tα by α, given by Ξ(Tα,m|[0,1))(n) =(1[1−α,1) ∗ ˜1[1−α,1))({αn}) for all n ∈ Z and ˆΞ(Tα,m|[0,1))(n) = | sin

(πn(1−α)

)/(πn)|2

for all n ∈ Z\{0}, ˆΞ(Tα,m|[0,1))(0) = α2 and ∥ˆΞ(Tα,m|[0,1))∥ = α.

Proof. The proof starts off in exactly the same way as the proof of Theorem 8.5.8,up to the point of (8.10), for which there exists an vm ∈ X given in terms ofCorollary 8.5.5 such that S sv(m) = σm(vm) for some s ∈ N. As S 2σm(0) =S 2κkm/nm

0 = S 2σm(1) = S 2κkm/nm1 as in Theorem 2.3.8 corresponds to the map

Tkm/nm(x) = {x + km/nm}, one has for any N ∈ N,

|S sv(m)|nmN |w

nmN=|σm(vm)|nmN |w

nmN

SN|S 2σm(0)|w

nmN± N

2|w|nmN

=N|S 2σm(0)|w

nmN±

2|w|nm

SN|σm(0)|w

nmN±

2|w| + 2nm

S|σm(0N)nmN |w

nmN±

4|w| + 2nm

.

In (8.9) and Theorem 2.3.8 we have seen that this is an approximation of the1[1−km/nm)∪{0}((1 − km/nm) + ·)-weighted autocorrelation of Tkm/nm . The coefficientsof the autocorrelation are exactly given by Ξ(Tm, µkm/nm) of (6.5), where µkm/nm =

1/nmδn−1m Znm

. Hence, for any ε > 0, w ∈ Lz−1(X), z ∈ N+ exists an M ∈ N such that ∑w∈Lz−1(X)

|S sv(m)|nmN |1w1

nmN− Ξ(Tkm,nm , µkm/nm)(z)

≤

∑w∈Lz−1(X)

|σm(0N)nmN |1w1

nmN− Ξ(Tkm,nm , µkm/nm)(z)

+ 2z−1 4(z + 1) + 2

nm

≤ ε + 2z−1 4(z + 1) + 2nm

,

for all m ≥ M. The remainder gets arbitrary small as m tends to infinity byassumption. Note that we have by translation invariance of ηkm/nm and definition


of (6.9), (6.10) for any w ∈ n−1m Znm

Ξ(Tkm,nm , µkm/nm,w)(z)= (1[1−km/nm,1)∪{0})km/nm,w ∗ ((1[1−km/nm,1)∪{0})km/nm,w)∼(z)

=1

nm

∫1[1−km/nm,1)∪{0}(x − km/nmz + w) · (1[1−km/nm,1)∪{0})∼(x + w) dδn−1

m Znm(x)

=1

nm

∫1[1−km/nm,1)∪{0}(x − km/nmz) · (1[1−km/nm,1)∪{0})∼(x) dδn−1

m Znm(x)

=

∫1[1−km/nm,1)(x − km/nmz) · (1[1−km/nm,1))∼(x) dm|[0,1)(x)

= 1[1−km/nm,1) ∗ (1[1−km/nm,1))∼(km/nmz) = Ξ(Tkm,nm ,m|[0,1))(z),

since∫1[0,1/nm) dm|[0,1) = n−1

m 1[0,1/nm)(0). Then pointwise convergence of the se-quence Ξ(Tkm,nm , ηkm/nm) to Ξ(Tα,m|[0,1)) can be derived from uniform convergenceof 1[1−km/nm,1) ∗ (1[1−km/nm,1))∼ to 1[1−α,1) ∗ (1[1−α,1))∼. Further note that in particular

ˆγTα =∑n∈Z

|(1[1−α,1))∧|2(n) δ{αn} =∑n∈Z

∫ 1

01[1−α,1)e−2πixn dm(x)

2 δ{αn}

=∑n∈Z

1 − e−2πin(1−α)

−2πin

2δ{αn} =

∑n∈Z

sin

(πn(1 − α)

)πn

2δ{αn},

by Theorems 6.3.7 and 6.3.10. �

Remark 8.5.10. As Lemma 6.3.8 and Theorems 6.3.7 and 6.3.10 were used at theend of the proof, the vague limit coincides with the one for maps x ↦→ {x + αm},i.e. βm = 1 for any m ∈ N. Hence we can assume (βm, αm) ∈ Dkm,nm ∪ {(1, km/nm)}in Proposition 8.5.9, where the closure is taken in R2.

In [83, Thm. 7.3.3] it were shown that within the setting of Proposition 8.5.9,the Parry measures µm converge vaguely to the Lebesgue measure m for m → ∞.With that, different weight-functions for the return time combs can be chosen,which is realised in the following proposition. The condition on the weight-functions in Proposition 8.5.11 is inspired by convergence in measure for func-tions, but needs an additional assumption which ensures they do not vary too largeon small neighbourhoods for too many points.

Proposition 8.5.11. For sequences((km, nm)

)m∈N with gcd(km, nm) = 1, let Dkm,nm

be associated with σm for all m ∈ N. Given ym ∈ [0, 1) and (βm, αm) ∈ Dkm,nm ,m ∈ N we define the map Tm(x) ≔ {βmx + αm} with Parry density µm, wherem ∈ N and bounded measurable functions f , fm : [0, 1) → C, where m ∈ N andsup{∥ fm∥∞ : m ∈ N} < ∞.


If limm→∞ km/nm = α and nm → ∞, the set of discontinuities of f has Lebesguemeasure zero and for all δ, ε1, ε2 > 0 there exists an M ∈ N such that

m

⎛⎜⎜⎜⎜⎜⎝⋃m≥M

{z ∈ [0, 1) : sup

x,y∈Bδ(z)| fm(y) − f (x)| ≥ ε2

}⎞⎟⎟⎟⎟⎟⎠ < ε1, (8.11)

the autocorrelations γTm of the fm-weighted return time combs with respect to theParry measure µm and reference point ym converge to

v-limm→∞ γTm = γTα ,

where γTα is the autocorrelation of a f -weighted return time comb with respect tom and Tα(x) = {x + α}.

Proof. As vague convergence for functionals on Z is the same as pointwise con-vergence of the corresponding functions and γTm(−n) = γTm(n) for all n ∈ Z weconsider a fixed n ∈ N. Set gm ≔ fm ◦ T n

m · fm, g ≔ f ◦ T nα · f . With that we have

|γTα(n) − γTm(n)| = |⟨m, g⟩ − ⟨µm, gm⟩|

= |⟨m, g⟩ − ⟨µm, gm + g − g⟩|= |⟨m, g⟩ − ⟨µm, g⟩ + ⟨µm, g − gm⟩|

≤ |⟨m, g⟩ − ⟨µm, g⟩| +∫|g − gm| hm dm. (8.12)

To continue we remark that the set of discontinuities of a function ϕ will be de-noted by Uϕ and for another function ψ it holds in general

Uϕ·ψ ⊆ Uϕ ∪ Uψ and Uϕ◦ψ ⊆ Uψ ∪ ψ−1(Uϕ). (8.13)

Convergence to zero of the first term is then exactly the vague convergence ofµm to m by [83, Thm. 7.3.3], together with Theorem 4.2.3, as U f has Lebesguemeasure zero and Tα is a homeomorphism the set Ug has Lebesgue measure zero.

Regarding the second term we first notice for T nm its supremum-norm ∥T n

m −

T nα∥∞ on ([0, 1),+) has, for large m, an upper bound via⎧⎪⎪⎨⎪⎪⎩

⎧⎪⎪⎨⎪⎪⎩βnmx + αm

n−1∑i=0

βim

⎫⎪⎪⎬⎪⎪⎭ − {x + nα}

⎫⎪⎪⎬⎪⎪⎭ =⎧⎪⎪⎨⎪⎪⎩(βn

mx − x) + (αm

n−1∑i=0

βim − nα)

⎫⎪⎪⎬⎪⎪⎭ ,which is smaller than 1/2 if (βn

m − 1) < 1/4 and (αm∑n−1

i=0 βim − nα) < 1/4. The

implied bound by taking the supremum with respect to x is clearly convergingto zero, as m tends to infinity. Hence there exists an M ∈ N such that for anyz ∈ [0, 1) we have |T n

m(z) − T nα(z)| < δ for all m ≥ M and thus

| fm(T nm(z)) − f (T n

α(z))| = supx,y∈Bδ(T n

m(z))| fm(x) − f (y)|


Moreover

|gm(z) − g(z)| = | fm ◦ T nm(z) · fm(z) − f ◦ Tα(x) · f (z)|

= | fm ◦ T nm(z) · ( fm(z) + f (z) − f (z)) − f ◦ Tα(z) · f (z)|

≤ | fm ◦ T nm(z) · fm(z) − fm ◦ T n

m(z) · f (z)|

+ | fm ◦ T nm(z) · f (z)) − f ◦ Tα(z) · f (z)|

≤ | fm ◦ T nm(z)| | fm(z) − f (z)| + | f (z)| | fm ◦ T n

m(z) − f ◦ Tα(z)|≤ C1 sup

x,y∈Bδ(T nm(z))| fm(x) − f (y)|,

for some C1 > 0, since sup{∥ fm∥∞ : m ∈ N} < ∞. With that we observe∫|g − gm| hm dm

=

∫{|gm−g|≤ε2}

|g − gm| hm dm +∫{|gm−g|>ε2}

|g − gm| hm dm

≤ ε2µm([0, 1)) + ∥(gm) − g∥∞ µm({|gm − g| > ε2})≤ ε2 1 + ∥(gm) − g∥∞ µm({ sup

x,y∈Bδ(T nm(z))| fm(x) − f (y)| > ε2/C1})

≤ ε2 +C2 µm

⎛⎜⎜⎜⎜⎜⎝⋃k≥M

{z ∈ [0, 1) : supx,y∈Bδ(T n

m(z))| fk(x) − f (y)| > ε2/C1}

⎞⎟⎟⎟⎟⎟⎠= ε2 +C2 µm

⎛⎜⎜⎜⎜⎜⎝T−nm

⋃k≥M

{z ∈ [0, 1) : supx,y∈Bδ(z)

| fk(x) − f (y)| > ε2/C1}

⎞⎟⎟⎟⎟⎟⎠= ε2 +C2 µm ◦ T−n

m

⎛⎜⎜⎜⎜⎜⎝⋃k≥M

{z ∈ [0, 1) : supx,y∈Bδ(z)

| fk(x) − f (y)| > ε2/C1}

⎞⎟⎟⎟⎟⎟⎠= ε2 +C2 µm

⎛⎜⎜⎜⎜⎜⎝⋃k≥M

{z ∈ [0, 1) : supx,y∈Bδ(z)

| fk(x) − f (y)| > ε2/C1}

⎞⎟⎟⎟⎟⎟⎠where we used that µm is Tm-invariant and µm(T−1

m (A)) = µm(T−1m (A)) for all closed

sets A, since T−1m (A) ⊆ T−1

m (A) ∪ UTm and µm(TTm) = 0 as µm ≪ m|[0,1) for anym ∈ N. Note also there exists a C2 > 0 such that ∥(gm) − g∥∞ < C2 < ∞, assup{∥ fm∥∞ : m ∈ N} < ∞. By Theorem 4.2.3, lim supm→∞ µm(A) ≤ m(A) for allclosed sets A ⊆ ([0, 1),+) and

m

⎛⎜⎜⎜⎜⎜⎝⋃k≥M

{z ∈ [0, 1) : supx,y∈Bδ(z)

| fk(x) − f (y)| > ε2/C1}

⎞⎟⎟⎟⎟⎟⎠ < ε1,

which finishes the proof, as ε1, ε2 > 0 can be chosen arbitrarily small. �


Next we give some examples of functions f , fm, m ∈ N such that condition(8.11) given in Proposition 8.5.11 is fulfilled.

Example 8.5.12. 1. Let ϕ ∈ C(([0, 1),+),C), then fm = ϕ for all m ∈ N andf = ϕ clearly satisfy (8.11).

2. Let f , fm be piecewise continuous bounded functions with finitely manydiscontinuities U f ,U fm ⊆ [0, 1), where m ∈ N and we further assumesup{|U fm | : m ∈ N} < ∞ and sup{∥ fm∥∞ : m ∈ N} < ∞. If fm → fpointwise and it exists a finite set U ⊆ [0, 1) such that dist(U fm ,U) → 0 form→ ∞, then (8.11) is also satisfied.

This is due to to the fact that it exists for all ε1, ε2 > 0 an M ∈ N such that⋃k≥M

U fm ⊆ U + (−ε1, ε1).

Then fm, f are uniformly continuous functions on [0, 1)\U + (−ε1, ε1) andconverge uniformly to f . Therefore for any ε2 it exists a δ > 0 such that| fm(y) − fm(x)| < ε2/2 for all x, y ∈ [0, 1) such that |x − y| < δ and m ∈ N.By choosing M such that ∥ fm − f ∥∞ < ε2/2 for all m ≥ M we have | fm(y) −f (x)| = | fm(y) − fm(x) + fm(x) − f (x)| ≤ | fm(y) − fm(x)| + | fm(x) − f (x)| < ε2

for all m ≥ M. Thus

m

⎛⎜⎜⎜⎜⎜⎝⋃m≥M

{ supx,y∈Bδ(z)

| fm(y) − f (x)| ≥ ε2}

⎞⎟⎟⎟⎟⎟⎠≤ m

⎛⎜⎜⎜⎜⎜⎝(U + (−ε1, ε1)) ∩⋃m≥M

{ supx,y∈Bδ(z)

| fm(y) − f (x)| ≥ ε2}

⎞⎟⎟⎟⎟⎟⎠+ m

(U + (−ε1, ε1)

)= card(U) · 2ε1,

which converges to zero, as ε1 tends to zero.

3. The characteristic functions 1[γn,1),1[γ,1), where γn → γ as n tends to in-finity satisfy (8.11), since they are a special class of piecewise continuousfunctions. These are chosen in Proposition 8.5.9 and thus one can also de-duce Proposition 8.5.9 from Proposition 8.5.11, even though the proofs usetechniques which are not related.

4. Let fm, f : [0, 1) → C be bounded and Lebesgue-measurable functions. Iflimm→∞ ∥ fm − f ∥∞ = 0, then (8.11) of Proposition 8.5.11 holds. Note thatthe condition ∥ fm∥∞ < C for all m ∈ N and some C > 0 is implicitly satisfiedby the uniformly convergence.


Remark 8.5.13. The assumptions on fm, f made in Theorem 6.3.10 are also ful-filled in Example 8.5.12(4) and thus Proposition 8.5.11 can be applied. This letus post an alternate version of Remark 8.5.10, which states that for return timecombs generated from functions of this kind with respect to transformations re-garding (βm, αm) ∈ Dkm,nm ∪ {1, km/nm}, we have convergence of the autocorrela-tions and spectral return measures and in case (βm, αm) = (1, km/nm) we chooseTheorem 6.3.10 and Proposition 8.5.11 otherwise.

Also note that a similar approach would work for functions with assumptionsmade in Example 8.5.12(2), where in this case the proof of Theorem 6.3.10 has tobe modified.

We close this section by noting down some observations for substitutions σ ∈Q. Note that σ ends either in ττTM, ρτTM or is equal to τTM. For the first two notethat they admit a coincidence (see Definition 6.6.14 for the definition) as

ττTM :

⎧⎪⎪⎨⎪⎪⎩0 ↦→ 0101 ↦→ 100

, ρτTM :

⎧⎪⎪⎨⎪⎪⎩0 ↦→ 0111 ↦→ 101

. (8.14)

Now we want to calculate the height of σ, see Definition 6.6.10. For that, firstsuppose σ = τ . . . τTM ∈ Q is of length q ∈ N. As τ(lθl)[0,1] = lθl, ρ(lθl)[0,1] = lθl,where l ∈ {0, 1} one has σ(1)|[0,2] = τ(10) = 100 and σ(0)|[0,2] = τ(01) = 010.Together with σn−1(l) being a prefix of σn(l) for l ∈ {0, 1} this yields

σ2(1)|[0,2] = 100, σ2(1)|[q,q+2] = 010, σ2(1)|[2q,2q+2] = 010, (8.15)

where there are 1’s at positions q + 1, 2q + 1 and gcd(q + 1, 2q + 1) = 1 for allq ∈ N, since any divisor s ≥ 2 of q + 1 also divides 2(q + 1) = 2q + 2 and hencecannot divide 2q + 1. Now suppose σ = ρ . . . τTM ∈ Q is of length q ∈ N. Thenσ(1)|[0,2] = ρ(10) = 101 and σ(0)|[0,2] = ρ(01) = 011 for l ∈ {0, 1} and

σ2(1)|[0,2] = 101, σ2(1)|[q,q+2] = 011, σ2(1)|[2q,2q+2] = 101,

where there are 1’s at positions q + 1, q + 2 and gcd(q + 1, q + 2) = 1 for allq ∈ N. Therefore, in both cases gcd(S 0) = 1, where S 0 is defined as in Defini-tion 6.6.10. It follows h(σ) = 1 and admits a coincidence, see (8.14), thus σ hasdiscrete spectrum by Theorem 6.6.16. The next proposition is an application ofProposition 6.6.18.

Proposition 8.5.14. Let m ∈ N and σi ∈ Q for all 0 ≤ i ≤ m, where it existsan 0 ≤ j ≤ m such that σ j ≠ τTM. For σ ≔ σ0 . . . σm of constant length q, thesubshift Xσ has discrete spectrum supported on the set

⋃n∈N+(q

−nZqn).

As an immediate consequence we have the following proposition. Here asequence

((ki, ni)

)i∈N is called periodic if there exists a p ∈ N+ such that ki+p = ki,

ni+p = ni for all i ∈ N


Proposition 8.5.15. Let ℓ ≔((ki, ni)

)i∈N with gcd(ki, ni) = 1 be a periodic se-

quence with period p and at least one tuple being not equal to (1, 2). Let Dki,ni beassociated with σi for all i ∈ N. Set ℓm ≔

((ki, ni)

)mi=0 and define for (βm, αm) ∈ Dℓm

the map Tm(x) ≔ {βmx + αm} with Tm-discontinuity γm. Then, for the autocorre-lations γTm of the 1[γm,1)-weighted return time combs with respect to the Parrymeasure µm and reference point ym, one has that v-limm→∞ γTm = γu, whereu = limm→∞ σ

m(1) for σ ≔ σ0 . . . σp of constant length q and Xu = Xσ. Further-more, ˆγu is a discrete measure on [0, 1), with its atoms inside the set

⋃n∈N+(q

−nZqn)and ∥ˆγu∥ = k0/n0.

The equality ∥ˆγu∥ = k0/n0 in the last theorem is due to ∥ˆγu∥ = γ(0), whichis given by the frequency of ones in u and thus is definitely given by the ratio|σ(1)|1/|σ(1)|. But since |σ0(1)|1 = |σ0(0)|1, the last substitution fully determinesthe frequency of ones.

8.5.1 Thue-Morse substitution

The fact that τTM ∈ Q is a special case with different behaviour than the otherelements of Q has been shown in (8.14). The substitution τTM is called Thue-Morse substitution, Remark 2.3.3, and is known to have singular spectrum, see[70, 5, 71, 4, 7] for some of the more recent literature about it. Here, a recap ofthe proof to show that the resulting spectral measures are singular to the Lebesguemeasure will be done for our setting by using techniques of the former mentionedliterature.

For reasons that will show up later we do not start of with the alphabet {0, 1},but let Σ ≔ {a, b} and hence τTM(a) = ab, τTM(b) = ba. Further, for w ∈ Σ∗, c ∈ Σone has |τTM(wc)| = 2|w| + 2 and (τTM(wc))2|w| = c, (τTM(wc))2|w|+1 = θc, whereθ(a) = b, θ(b) = a. It is immediate to see that τn

TM(c)|2n−1 = τn−1TM (c) for c ∈ Σ and

n ∈ N+. This let us define

v ≔ limn→∞

τnTM(b),

and thus

v2n =vn, v2n+1 = θvn, n ∈ N (8.16)

Which is known as 2-periodicity. In the remainder of this section the return spec-trum of τTM, see Definition 6.6.8, will be calculated for the alphabets {−1, 1} and{0, 1}. To circumvent a reassignment of the alphabet we consider a function fwhich assigns a real value to each element of Σ. It can be extended to a functionon XτTM , by (ui)i∈N ↦→ f (u0). This function is continuous, as it is constant on every


cylinder. Set wi ≔ f (vi) for all i ∈ N. With that assignment the autocorrelation γv

for the return time comb µv ≔∑

i∈N wiδi is unique and one has from (6.2)

γv(m) = limN→∞

12N

2N−1∑i=0

wiwi+m

and γv(−m) = γv(m) for m ≥ 0. By 2-periodicity one yields

γv(2m) = limN→∞

12N

2N−1∑i=0

wiwi+2m

= limN→∞

12N

⎛⎜⎜⎜⎜⎜⎜⎝2N−1−1∑i=0

w2iw2i+2m +

2N−1−1∑i=0

w2i+1w2i+1+2m

⎞⎟⎟⎟⎟⎟⎟⎠= lim

N→∞

12N

⎛⎜⎜⎜⎜⎜⎜⎝2N−1−1∑i=0

wiwi+m +

2N−1−1∑i=0

θwiθwi+m

⎞⎟⎟⎟⎟⎟⎟⎠ (8.17)

γv(2m + 1) = limN→∞

12N

2N−1∑i=0

wiwi+2m+1

= limN→∞

12N

⎛⎜⎜⎜⎜⎜⎜⎝2N−1−1∑i=0

w2iw2i+2m+1 +

2N−1−1∑i=0

w2i+1w2i+2m+2

⎞⎟⎟⎟⎟⎟⎟⎠= lim

N→∞

12N

⎛⎜⎜⎜⎜⎜⎜⎝2N−1−1∑i=0

wiθwi+m +

2N−1−1∑i=0

θwiwi+m+1

⎞⎟⎟⎟⎟⎟⎟⎠ (8.18)

If f (a) = −1, f (b) = 1, then

γv(2m)(8.17)= γv(m), γv(2m + 1)

(8.18)= −1/2(γv(m) + γv(m + 1)). (8.19)

The values can now be computed inductively starting at m = 0 by γv(0) = γv(0)⇔γv(0) = 1. For the spectral (return) measure ˆγv first observe

µv|2N+1 =

2N+1∑i=0

wi δi =

2N∑n=0

w2i δ2i +

2N∑i=0

w2i+1 δ2i+1

(8.16)=

2N∑i=0

wi δ2i +

2N∑i=0

θwi δ2i+1.

(8.20)

hence for wN ≔ w|[0,N], N ∈ N

ˆw2N+1(x) =2N∑

n=0

wn e2πi2nx +

2N∑n=0

θwn e2πi(2n+1)x =

2N∑n=0

wn e2πin2x − e2πix2N∑

n=0

wn e2πin2x


=(1 − e2πix

) 2N∑n=0

wn e2πin2x =(1 − e2πix

) ˆw2N (2x),

where x ∈ [0, 1). This again admits a recursive representation starting withˆw20(x) = w0 e−2πi0x = 1 and as µv|2N+1 ∗ ˜µv|2N+1 can be used to approximate ˆγv

we take ˆw2N+1(x)2=

1 − e2πix

2 ˆw2N (2x)=

1 − eπi21 x

2 1 − e2πi22 x

2 ˆw2N−1(22x)

=

N+1∏n=1

1 − eπi2n x

2=

N+1∏n=1

2 (1 − cos(π2nx)) = 2N+1N+1∏n=1

(1 − cos(π2nx)) ,

where we have used that1 − e−2πi2N x

2=1 − e−2πi2N x − e2πi2N x + 1 = 2 − 2

e−2πi2N x + e2πi2N x

2=2

(1 − cos(π2N+1x)

).

These are the Lebesgue densities that converge to ˆγv in the vague limit and areknown as Riesz-products, [71, Ch. 1.3].

ˆγv=v-limN→∞

2−(N+1)(µv|2N+1 ∗ ˜µv|2N+1

)∧= v-lim

N→∞2−(N+1)

ˆw2N+1

2m|[0,1)

=v-limN→∞

2−(N+1)2N+1N+1∏n=1

(1 − cos(π2n·)) m|[0,1) = v-limN→∞

N+1∏n=1

(1 − cos(π2n·)) m|[0,1),

where m|[0,1) denotes the Lebesgue measure. The transform ˆγ is known to bemutually singular to the Lebesgue measure, [45, §7, 1] or [7, Ex. 10.1]. Forf (a) = 0 and f (b) = 1 one has from (8.17) and (8.18) for the autocorrelation γ′v

γ′v(2m) = limN→∞

12N

⎛⎜⎜⎜⎜⎜⎜⎝2N−1−1∑i=0

wiwi+m +

2N−1−1∑i=0

(1 − wi)(1 − wi+m)

⎞⎟⎟⎟⎟⎟⎟⎠= lim

N→∞

12N

⎛⎜⎜⎜⎜⎜⎜⎝2N−1−1∑i=0

wiwi+m +

2N−1−1∑i=0

1 − wi − wi+m + wiwi+m

⎞⎟⎟⎟⎟⎟⎟⎠= lim

N→∞

12N

⎛⎜⎜⎜⎜⎜⎜⎝2N−1−1∑i=0

wiwi+m +

2N−1−1∑i=0

wiwi+m

⎞⎟⎟⎟⎟⎟⎟⎠ + 12N

2N−1−1∑i=0

1 − wi − wi+m

=γ′v(m) + 1/2 − 1/4 − 1/4 = γ′v(m)

γ′v(2m + 1) = limN→∞

12N

⎛⎜⎜⎜⎜⎜⎜⎝2N−1−1∑i=0

wi(1 − wi+m) +2N−1−1∑

i=0

(1 − wi)wi+m+1

⎞⎟⎟⎟⎟⎟⎟⎠


= limN→∞

12N

⎛⎜⎜⎜⎜⎜⎜⎝2N−1−1∑i=0

wi −

2N−1−1∑i=0

wiwi+m +

2N−1−1∑i=0

wi+m+1 −

2N−1−1∑i=0

wiwi+m+1

⎞⎟⎟⎟⎟⎟⎟⎠=1/4 − 1/2γ′v(m) + 1/4 − 1/2γ′v(m + 1)=1/2(1 − γ′v(m) − γ′v(m + 1)).

Set η(m) ≔ 4γ′v − 1, then

η(2m) =4γ′v(2m) − 1 = 4γ′v(m) − 1 = η(m),η(2m + 1) =4γ′v(2m + 1) − 1 = 4/2(1 − γ′v(m) − γ′v(m + 1)) − 1

=4/2(1 − (η(m)/4 + 1/4) − (η(m + 1) + 1/4)

)− 1

=1/2(−η(m) − η(m + 1)) + 4/4 − 1= − 1/2(η(m) + η(m + 1))

A comparison with (8.19) yields that the formulas for η and γv coincide and asγv(0) = 1 and η(0) = 4γ′v(0) − 1 = 4/2 − 1 = 1 we have

γ′v = 1/4γv + 1/4 δZ, ˆγ′v = 1/4ˆγv + 1/4 δ0.

Proposition 8.5.16. Let v = limn→∞ τnTM(b) the limit of the Thue-Morse substitu-

tion in {a, b}N and set w ≔ f−1,1(v), w′ ≔ f0,1(v), where fx,y(c) is x if c = a andy if c = b. In this case the autocorrelations γw, γw′ exist and γw′ = 1/4γw +

1/4δZ. Moreover ˆγw′ = 1/4ˆγw + 1/4 δ0, where ϱ ≔ ˆγw denotes the Thue-Morse measure on [0, 1], mutually singular to the Lebesgue measure given byv-limN→∞

∏N+1n=1 (1 − cos(π2n·)) m|[0,1).

8.5.2 Variation of the Thue-Morse caseLet us denote by v the sequence associated with the Thue-Morse substitution andby γv the autocorrelation of v. Denote by v(q) the sequence given by

v(q)n ≔

⎧⎪⎪⎨⎪⎪⎩vn/q, n/q ∈ Z0, otherwise

.

Lemma 8.5.17. For sequences w,w′ ∈ ΣN ⊂ RN such that w ~ ˜w′ exists one has

w(q) ~ ˜w′(q)(qm + s) = q−1(w ~ ˜w)(q)(qm + s) =

⎧⎪⎪⎨⎪⎪⎩0 , 1 ≤ s ≤ q − 1q−1w ~ ˜w(m) , s = 0

where (w ~ ˜w)(q) = (w ~ ˜w)(q,0), see Definition 8.4.4.


Proof. Let m < 0. For all 1 ≤ s ≤ q − 1 by Corollary 6.3.5

w(q) ~ ˜w′(q)(qm + s) = limN→∞

1N

N−1∑i=0

w(q)i w′(q)

i+q|m|−s = 0

and for s = 0,

w(q) ~ ˜w′(q)(qm) = limN→∞

1qN

qN−1∑i=0

w(q)i w′(q)

i+q|m|

= limN→∞

1qN

N−1∑i=0

w(q)qi w′(q)

qi+q|m|

=1q

limN→∞

1N

N−1∑i=0

wiw′i+|m| = q−1w ~ ˜w′(m).

The case m ≥ 0 follows in analogously and the equality to (w ~ ˜w)(q) is thenimmediate by a comparison of both measures. �

With that 2−1⟨v~˜w, f ◦ q⟩ = ⟨v(q) ~˜w(q), f ⟩ for all f ∈ Cc(Z), where f ◦ q(z) ≔f (qz) for all z ∈ Z.

Lemma 8.5.18. The transform of (v(q) ~ ˜v(q)) is a sum of non-negative measures,mutually singular to the Lebesgue-measure, given by

(v(q) ~ ˜v(q))∧ = (2q)−2(ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq+ (2q)−2δ 1

qZq

where ϱ ≔ ˆγv on [0, 1] denotes the spectral return measure of the Thue-Morsesubstitution, also given by ϱ = v-limN→∞

∏N+1n=1 (1 − cos(π2n·)) m|[0,1). and ϱ◦ s−1

1/q,0is ϱ transported to [0, 1/q] by multiplication with q.

Proof. By Proposition 8.5.16 we have (v ~˜v)∧ = ϱ + 1/4δ0. The proof is a con-secutive application of Lemma 8.5.17 and Lemma 8.4.5

(v(q) ~ ˜v(q))∧ = q−1((v ~˜v)(q,0))∧

= q−2(e0(v ~˜v∧) ◦ s−1

1/q,0

)∗ (e0δ 1

qZq)

= q−2((1/4ϱ + 1/4δ0) ◦ s−1

1/q,0

)∗ δ 1

qZq

= (2q)−2(ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq+ (2q)−2δ 1

qZq.

That ϱ ◦ s−11/q,0 is still mutually singular to the Lebesgue measure is due to m being

equivalent to m◦s−11/q,0, by m◦s−1

1/q,0(A) = m(sq,0A) = qm(A) for all A ⊆ [0, 1/q]. �


Theorem 8.5.19. Let gcd(k, q) = 1 and u ≔ limm→∞ στmTM(1) for a σ associated

with Dk,q ≠ D1,2, then

(u ~˜u)∧ = 2−1q−2(1 − g1)(ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq+ φw,q δ 1

qZq,

where ϱ denotes the Bochner transform of the Thue-Morse substitution, as givenin Lemma 8.5.18 and g1(x) ≔ cos(2πx). The density φw,q is given by

φw,q ≔1q2

1/2 + e1/2 +

∑0≤r≤q−1,(00w)r=1

er

2 δq−1Zq ,

where w ≔ S 2σ(0) = S 2σ(1) and ey(x) = e2πixy. In particular ∥(u ~˜u)∧∥ = k/q.

There will be two proofs of Theorem 8.5.19 of which the second one is of amore informal nature, but gives a better revelation of the techniques that are used.

Proof of Theorem 8.5.19. That ∥(u~˜u)∧∥ = k/q is a direct consequence of the factthat |σ(0)|1 = |σ(1)|1 = k and |σ(0)| = |σ(1)| = q. Thus the frequency of 1’s is uis exactly k/q, hence ∥(u ~ ˜u)∧∥ = u ~ ˜u(0) = k/q. Let v = limm→∞ τ

mTM(1),

then u = σ(v) = σ′τTM(v), where σ′τTM ∈ Q and due to Corollary 2.3.6 itexists a word w ∈ {0, 1}q−2 such that S 2σ(0) = S 2σ(1) = w. As v is a fixedpoint of τTM one has u = v0v1wv2v3wv4v5w . . . and by (8.16) one deduces u =v0(θv0)wv1θ(v1)wv2θ(v2)w . . .. This leads to the following decomposition of u

u = v(q) + V + (00w)N, (8.21)

where Vi+1 ≔ (θv(q))i for all i ∈ N and V0 ≔ 0 and especially S (V) = θv(q). Withthat the autocorrelation of u decomposes into

u ~˜u = v(q) ~ ˜v(q) + v(q) ~ ˜V + V ~ ˜v(q)

+ V ~ ˜V + v(q) ~ ˜(00w)N + (00w)N ~ ˜v(q)

+ (00w)N ~ (00w)N + V ~ ˜(00w)N + (00w)N ~ ˜V . (8.22)

We will handle each part separately. For m < 0, 0 ≤ s ≤ q − 1

v(q) ~ ˜V(qm + s) = limN→∞

1N

N−1∑i=0

v(q)i (θv(q))i+q|m|−s−1

= limN→∞

1qN

qN−1∑i=0

v(q)i (θv(q))i+q|m|−s−1

=

⎧⎪⎪⎨⎪⎪⎩ 1q limN→∞

1N

∑N−1i=0 vi (θv)i+|m+1| , s = q − 1

0 , otherwise.


For m ≥ 0, 0 ≤ s ≤ q − 1

v(q) ~ ˜V(qm + s) = limN→∞

1qN

qN−1∑i=1

v(q)i+qm+s (θv(q))i−1

=

⎧⎪⎪⎨⎪⎪⎩ 1q limN→∞

1N

∑N−1i=1 vi+m+1 (θv)i , s = q − 1

0 , otherwise.

In a similar fashion for m < 0, 0 ≤ s ≤ q − 1

V ~ ˜v(q)(qm + s) = limN→∞

1qN

qN−1∑i=1

(θv(q))i−1 v(q)i+q|m|−s

=

⎧⎪⎪⎨⎪⎪⎩ 1q limN→∞

1N

∑N−1i=1 (θv)i vi+|m| , s = 1

0 , otherwise

and for m ≥ 0, 0 ≤ s ≤ q − 1

V ~ ˜v(q)(qm + s) = limN→∞

1qN

qN−1∑i=1

(θv(q))i+qm+s−1 v(q)i

=

⎧⎪⎪⎨⎪⎪⎩ 1q limN→∞

1N

∑N−1i=1 (θv)i+m vi , s = 1

0 , otherwise.

Combining the former two yields

(v(q) ~ ˜V + V ~ ˜v(q))(qm + s) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩1q limN→∞

1N

∑N−1i=1 (θv)i+|m| vi , s = 1

1q limN→∞

1N

∑N−1i=1 (θv)i vi+|m+1| , s = q − 1

0 , otherwise

=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩1/q(1/2 − γv(m)) , s = 11/q(1/2 − γv(m + 1)) , s = q − 10 , otherwise

for all m ∈ Z, 0 ≤ s ≤ q − 1. Notice for all m ∈ Z, 0 ≤ s ≤ q − 1

v(q) ~ ˜v(q) ◦ (+1)(qm + s) =

⎧⎪⎪⎨⎪⎪⎩v(q) ~ ˜v(q)(qm + s + 1) , s ≤ q − 1v(q) ~ ˜v(q)(q(m + 1)) , s = q − 1

=

⎧⎪⎪⎨⎪⎪⎩0 , otherwise(1/q) v ~˜v(m + 1) , s = q − 1


by Lemma 8.5.17 and in the same manner

v(q) ~ ˜v(q) ◦ (−1)(qm + s) =

⎧⎪⎪⎨⎪⎪⎩v(q) ~ ˜v(q)(qm + s − 1) , s ≥ 1v(q) ~ ˜v(q)(q(m − 1) + q − 1)) , s = 0

=

⎧⎪⎪⎨⎪⎪⎩(1/q) v ~˜v(m) , s = 10 , otherwise

As γv = v ~˜v, we have(v(q) ~ ˜V + V ~ ˜v(q)) = (2q)−1δqZ+1 + (2q)−1δqZ−1

− v(q) ~ ˜v(q) ◦ s1,1 − v(q) ~ ˜v(q) ◦ s1,−1

By linearity of the Bochner transform and Lemma C.3.2, Lemma 8.5.18 andProposition C.3.1 one has(

v(q) ~ ˜V + V ~ ˜v(q))∧= (2q)−1q−1e−1δ 1

qZq+ (2q)−1q−1e1δ 1

qZq

− e1

((2q)−2

(ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq+ (2q)−2δ 1

qZq

)− e−1

((2q)−2

(ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq+ (2q)−2δ 1

qZq

)= q−2(e1 + e−1)

(2−1δ 1

qZq−

( (2−2ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq+ 2−2δ 1

qZq

))= 2q−2g1

(4−1δ 1

qZq−

(2−2ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq

)(8.23)

= 2−1q−2g1

(δ 1

qZq−

(ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq

). (8.24)

In the next case one has for m < 0 and 0 ≤ s ≤ q − 1

v(q) ~ ˜(00w)N(qm + s) = limN→∞

1qN

qN−1∑i=0

v(q)i (00w)Ni+q|m|−s

=1q

limN→∞

1N

N−1∑i=0

vi(0w0)q−1−s

=

⎧⎪⎪⎨⎪⎪⎩1/(2q) , (0w0)q−1−s = 10 , (0w0)q−1−s = 0

=∑

0≤r≤q−1,(0w0)q−1−r=1

12qδqZ+r(qm + s),


as the number of ones and zeros in (vi)i∈N, the Thue-Morse sequence, have density1/2. In a similar manner for m ≥ 0, 0 ≤ s ≤ q − 1

v(q) ~ ˜(00w)N(qm + s) =

⎧⎪⎪⎨⎪⎪⎩1/(2q) , (0w0)q−s−1 = 10 , otherwise

=∑

0≤r≤q−1,(0w0)q−1−r=1

12qδqZ+r(qm + s).

In the same way

(00w)N ~ ˜v(q)(qm + s) = limN→∞

1qN

i=0∑qN−1

(00w)Ni v(q)i+q|m|−s

= limN→∞

1qN

i=0∑N−1

(00w)svi

=

⎧⎪⎪⎨⎪⎪⎩1/(2q) , (00w)s = 10 , (00w)s = 0

where m < 0, 0 ≤ s ≤ q − 1 and for m ≥ 0, 0 ≤ s ≤ q − 1

(00w)N ~ ˜v(q)(qm + s) = limN→∞

1qN

i=0∑qN−1

(00w)Ni+qm+sv(q)i

= limN→∞

1qN

i=0∑N−1

(00w)svi

=

⎧⎪⎪⎨⎪⎪⎩1/(2q) , (00w)s = 10 , (00w)s = 0

.

With that

v(q) ~ ˜(00w)N + (00w)N ~ ˜v(q)

=∑

0≤r≤q−1,(0w0)q−1−r=1

12qδqZ+q−1−r +

∑0≤r≤q−1,(00w)r=1

12qδqZ+r

=∑

0≤r≤q−1,(00w)r=1

12qδqZ−r +

∑0≤r≤q−1,(00w)r=1

12qδqZ+r. (8.25)

In the following case one has for m < 0 and 0 ≤ s ≤ q − 1

V ~ ˜(00w)N(qm + s) = limN→∞

1qN

qN−1∑i=1

(θv(q))i−1 (00w)Ni+q|m|−s


= limN→∞

N−1∑i=0

(θv)i(w00)q−1−s =1

2q

∑0≤r≤q−1,(0w0)r=1

δqZ−r(qm + s)

and by the same arguments as before for m ≥ 0, 0 ≤ s ≤ q − 1

V ~ ˜(00w)N(qm + s) = limN→∞

1qN

qN−1∑i=1

(θv(q))i+qm+s−1 (00w)Ni

= limN→∞

N−1∑i=0

(θv)i+m(w00)q−1−s

=1

2q

∑0≤r≤q−1,(0w0)r=1

δqZ−r(qm + s).

The next case has for m < 0, 0 ≤ s ≤ q − 1

(00w)N ~ ˜V(qm + s) = limN→∞

1qN

qN−1∑i=1

(00w)Ni (θv(q))i−1+q|m|−s

= limN→∞

1qN

N−1∑i=1

(0w0)s (θv(q))i

=1

2q

∑0≤r≤q−1,(0w0)r=1

δqZ+r(qm + s).

and for m ≥ 0, 0 ≤ s ≤ q − 1

(00w)N ~ ˜V(qm + s) = limN→∞

1qN

qN−1∑i=1

(00w)Ni+qm+s (θv(q))i−1

= limN→∞

1qN

N−1∑i=1

(0w0)s (θv(q))i

=1

2q

∑0≤r≤q−1,(0w0)r=1

δqZ+r(qm + s).

With that

V ~ ˜(00w)N + (00w)N ~ ˜V = 12q

∑0≤r≤q−1,(0w0)r=1

δqZ−r +12q

∑0≤r≤q−1,(0w0)r=1

δqZ+r.

and together with (8.25) it yields(v(q) ~ ˜(00w)N + (00w)N ~ ˜v(q) + V ~ ˜(00w)N + (00w)N ~ ˜V)∧


=

⎛⎜⎜⎜⎜⎜⎜⎝ ∑0≤r≤q−1,(00w)r=1

12qδqZ−r +

∑0≤r≤q−1,(00w)r=1

12qδqZ+r

+1

2q

∑0≤r≤q−1,(0w0)r=1

δqZ−r +1

2q

∑0≤r≤q−1,(0w0)r=1

δqZ+r

⎞⎟⎟⎟⎟⎟⎟⎠∧

=1

2q2

⎛⎜⎜⎜⎜⎜⎜⎝ ∑0≤r≤q−1,(00w)r=1

erδ 1qZq+

∑0≤r≤q−1,(00w)r=1

e−rδ 1qZq

+∑

0≤r≤q−1,(0w0)r=1

erδ 1qZq+

∑0≤r≤q−1,(0w0)r=1

e−rδ 1qZq

⎞⎟⎟⎟⎟⎟⎟⎠=

1q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∑

0≤r≤q−1,(00w)r=1

grδ 1qZq+

∑0≤r≤q−1,(0w0)r=1

grδ 1qZq

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=

1q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∑

0≤r≤q−1,(00w)r=1

(gr + gr−1)δ 1qZq

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (8.26)

by Proposition C.3.1. Again, for m < 0, 0 ≤ s ≤ q − 1 one has

V ~ ˜V(qm + s) =

⎧⎪⎪⎨⎪⎪⎩limN→∞1

qN

∑N−1i=0 (θv)i (θv)i+|m| , s = 0

0 , otherwise

=

⎧⎪⎪⎨⎪⎪⎩(1/q)(1 − 1/2 − 1/2 + v ~˜v(m)) , s = 00 , otherwise

= v(q) ~ ˜v(q)(qm + s) (8.27)

and the same result is yielded for m ≥, 0 ≤ s ≤ q − 1. For m < 0, 0 ≤ s ≤ q − 1,the last case in this proof is

(00w)N ~ ˜(00w)N(qm + s) = limN→∞

1N

N−1∑i=0

(00w)Ni (00w)Ni+q|m|+s

= limN→∞

1qN

Nq−1∑i=0

(00w)i(00w)i+s mod q

= (00w) ∗ ˜(00w)(s),

where the convolution is taken with respect to Zq and has its Bochner transform


given by

((00w)N ~ ˜(00w)N

)∧=

1q2

∑0≤r≤q−1,(00w)r=1

er

2 δ 1

qZq.

Taking also Lemma 8.5.18 and (8.24), (8.26), (8.27) into account we have for(8.22) by linearity of the Bochner transform

(u ~˜u)∧ =(v(q) ~ ˜v(q) + V ~ ˜V)∧

+(v(q) ~ ˜V + V ~ ˜v(q))∧ (8.28)

+ v(q) ~ ˜(00w)N + (00w)N ~ ˜v(q) (8.29)

+ V ~ ˜(00w)N + (00w)N ~ ˜V)∧+

((00w)N ~ (00w)N

)∧= 2

((2q)−2

(ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq+ (2q)−2δ 1

qZq

)(8.30)

+ 2−1q−2g1

(δ 1

qZq−

(ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq

)

+1q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∑

0≤r≤q−1,(00w)r=1

(gr + gr−1)δ 1qZq

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ + 1q2

∑0≤r≤q−1,(00w)r=1

er

2 δ 1

qZq

= 2−1q−2(1 − g1)(ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq+ 2−1q−2

(g1δ 1

qZq+ 2−2δ 1

qZq

)

+1q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∑

0≤r≤q−1,(00w)r=1

(gr + gr−1) +

∑0≤r≤q−1,(00w)r=1

er

2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ δ 1

qZq

= 2−1q−2(1 − g1)(ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq+ q−2(1/2 + 2−1g1) δ 1

qZq

+1q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∑

0≤r≤q−1,(00w)r=1

(gr + gr−1) +

∑0≤r≤q−1,(00w)r=1

er

2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ δ 1

qZq. (8.31)

A closer look at the discrete terms in (8.31) reveals that

(1/2 + 2−1g1) = 1/4 + 1/4(e1 + e−1) + 1/4 = |1/2e0 + 1/2e1|2,

and we also have∑0≤r≤q−1,(00w)r=1

(gr + gr−1) =∑

0≤r≤q−1,(00w)r=1

er + e−r

2+

e1−r + er−1

2

=12

∑0≤r≤q−1,(00w)r=1

e0er + e0e−r + e1e−r + e−1er


=12

∑0≤r≤q−1,(00w)r=1

(e0 + e1)er + (e0 + e1)er

=∑

0≤r≤q−1,(00w)r=1

(1/2e0 + 1/2e1)er + (1/2e0 + 1/2e1)er.

Hence the discrete part in (8.31) can be written as

|1/2e0 + 1/2e1|2 + (1/2e0 + 1/2e1)

∑0≤r≤q−1,(00w)r=1

er

+ (1/2e0 + 1/2e1)∑

0≤r≤q−1,(00w)r=1

er +

∑0≤r≤q−1,(00w)r=1

er

2

=

12e0 +

12

e1 +∑

0≤r≤q−1,(00w)r=1

er

2 ,

which concludes the proof. �

Remark 8.5.20. Let n ∈ N, define u ≔ limm→∞ σ0 . . . σnτmTM(1) for substitutions

σ0, . . . , σn ∈ Q and set q ≔ |σ0 . . . σn(l)|, where l ∈ {0, 1}. Then the decomposi-tion of u in terms of (8.21) is more complicated, since, instead of only 00w, thereare two choices, either w0 or w1 for the latter part, depended on σ0 . . . σn beingapplied on either 0 or 1. With that it is conjectured that there are several copies ofit with factors er in front which do not completely cancel out.

8.6 Convergence of β-transformationsIn this section the results obtained in the previous sections of this chapter are col-lected, which concern limits for sequences of β-transformations associated with(βm, αm) ∈ ∆, m ∈ N approaching (1, α), where 0 < α < 1.

For that define Tm(x) ≔ {βmx + αm} for each x ∈ [0, 1) and m ∈ N. Wechoose the Parry density hm associated with every dynamical system for a β-transformation Tm and derive from that the autocorrelation γTm , which is for µm ≔hmm[0,1)-almost every y given by the 1[γm,1)-weighted return time comb with ref-erence point y, where m ∈ N and γm denotes the Tm-discontinuity of Tm. It is ingeneral hard to determine whether the vague limit (γm)m∈N for m→ ∞ even exists,but if (βm, αm) are in certain subsets of ∆ for every m ∈ N the existence of a vaguelimit can be guaranteed by combinatorial arguments. Namely we at least require(βm, αm) ∈ Dℓm for all m ∈ N, where Dℓm is a small area in ∆ given by a finite

8.6. CONVERGENCE OF β-TRANSFORMATIONS 167

sequence ℓm and the number of elements in each ℓm tend to infinity for m → ∞,see Definition 8.4.7.

If (Dℓm)m∈N is a nested sequence, i.e. Dℓm+1 ⊆ Dℓm for all m ∈ N, thenv-limm→∞ γTm = γu by Theorem 8.5.8, where u = limm→∞ σ0 . . . σm(1) ∈ {0, 1}N

generates a minimal subshift Xu ⊆ {0, 1}N, see Lemma 8.5.7 and each σi is as-sociated with a Dki,ni of which (ki, ni) is an element of the sequence ℓm for any0 ≤ i ≤ m. Thus v-limm→∞ ˆγTm = ˆγu by continuity of the Bochner transform,see [13, Thm. 3.13] and there are some cases in which ˆγu is known. One caseis if substitutions σi ∈ Q are chosen periodically with period n ∈ N. In thiscase define the substitution σ = σ0 . . . σn of constant length q such that Xu = Xσ

and if σ ≠ τTM, then ˆγu is a discrete measure supported on the set⋃

n∈N+(q−nZqn)

by Proposition 8.5.15. In the case σ = τTM associated with D1,2, the subshiftXu = XτTM and ˆγu has its only atom at 0 and is otherwise mutually singular to theLebesgue measure with the ever-popular vague limit of the Riesz products asso-ciated with the Thue-Morse sequence, see Proposition 8.5.16. The non-discrete,mutually singular to the Lebesgue measure part also appears if the first substitu-tion differs, i.e. u = limm→∞ σ0τ

mTM, where σ0 is associated with Dk,q and can, in

this case, be calculated explicitly to be

ˆγu = ϕq

(ϱ ◦ s−1

1/q,0

)∗ δ 1

qZq+ φσ0 δ 1

qZq,

where s1/q,0(x) ≔ x/q for all x ∈ [0, 1) and some densities ϕq, φσ0 ≠ 0 given inTheorem 8.5.19, which also provides the whole statement. If σ0 ∈ Q it is not hardto see that there are still non-discrete density measures mutually singular to theLebesgue measure, see Remark 8.5.20, but it has to be shown that their densitiesdo not cancel each other, which is conjectured to be the case.

Even if (Dℓm)m∈N is not a nested sequence, there is a case in which the existenceof the vague limit can be guaranteed. To achieve this, fix a sequence ((km, nm))m∈N

with nm → ∞ for m → ∞ and choose a Tm for every set Dkm,nm , m ∈ N. If thereexists an α ∈ [0, 1) such that limm→∞ km/nm = α, then

v-limm→∞ γTm = γTα and ˆγTα =∑n∈Z

sin

(πn(1 − α)

)πn

2 δ{αn}.

where γTα(n) = 1[1−α,1) ∗ 1[1−α,1)({αn}) for all n ∈ Z is the autocorrelation ofrotation by α for the 1[1−α,1)-weighted return time comb, see Proposition 8.5.9.This result has been generalised in Proposition 8.5.11 from 1[γm,1)-weighted returntime combs to fm-weighted return time combs for certain functions fm, whichconverge to a function f such that for all δ, ε1, ε2 > 0

m

⎛⎜⎜⎜⎜⎜⎝⋃m≥M

{z ∈ [0, 1) : sup

x,y∈Bδ(z)| fm(y) − f (x)| ≥ ε2

}⎞⎟⎟⎟⎟⎟⎠ < ε1


and the set U f of discontinuities of f has Lebesgue-measure zero. This is forexample fulfilled by piecewise continuous functions, with a uniform upper boundas described in Example 8.5.12(2). As another example one can choose fm tobe Riemann-integrable and non-negative with limm→∞ ∥ fm − f ∥∞ = 0. In thiscase one can even consider (βm, αm) ∈ Dkm,nm ∪ {1, km/nm}, m ∈ N as depicted inRemark 8.5.13.

Appendix A

Dynamical systems

A.1 ConjugaciesThis chapter uses techniques and definitions from [70, Ch. 1], [85, Ch. 2], [44,Ch. 2].

As with all disciplines of mathematics a great interest lies upon systems whichshare the same properties. For topological spaces X1, X2 that is given, if there isa homeomorphism h : X1 → X2. Two topological dynamical systems (X1,T1),(X2,T2) are said to be (topologically) conjugate if there exists a homeomorphismh : X1 → X2 such that h ◦ T1 = T2 ◦ h. If h is only a continuous surjection, then(X2,T2) is a (topological) factor of (X1,T1). In the case h is a measurable map,where (X1,B1,T1, µ1) is a Borel measure space and (X2,B2,T2) a measurablespace the measure µ1 can be transported onto X2 via h and in the case T1,T2

commute with h

µ1(T−11 (h−1A) △ X1) = µ1(h−1 ◦ T−1

2 (A) △ h−1(X2)) = µ1 ◦ h−1(T−12 (A) △ X2). (A.1)

Hence, one deduces that ergodicity for the transported measure is preserved un-der (topological) factors h. Similar calculations also shows these for strong andweakly mixing. If, additionally, (X2,B2,T2, µ2) is a measure space it is (measuretheoretically) isomorphic to (X1,B,T1, µ1), if there is a bijection h, measurablein both directions, between two sets Y1 ∈ B1, Y2 ∈ B2 of full measure such thath ◦ T1 = T2 ◦ h and µ1(h−1(A)) = µ2(A) for any A ∈ B1. The map h is then calleda conjugacy map between T1 and T2 and in the case h is only surjective it is saidto be a measure theoretic factor.

Example A.1.1 (Topologically conjugate). The map Tα : [0, 1) → [0, 1) given byx ↦→ {x + α} is topologically conjugate to the rotation map z ↦→ e2πiαz on the unitcircle S1 in C, where the conjugate is given by h : [0, 1)→ S1, x ↦→ e2πix.

169

170 APPENDIX A. DYNAMICAL SYSTEMS

Example A.1.2 (Measure theoretically isomorphic). Consider X1 = [0, 1] andX2 ≔ S

1 = {x ∈ C : ∥x∥ = 1}. Set h : [0, 1] → S1, by x ↦→ e2πix and h′ ≔h−1|[0,1) : S1 → [0, 1). Let us consider T1 : [0, 1] ↦→ [0, 1] and in the case T1(0) =

T1(1) we define a second map T2 by T2 ≔ h ◦ T1 ◦ h′ : S1 → S1. Then (S1,T2) is atopological factor of ([0, 1],T1) via h, as T2◦h(x) = h◦T1◦h′◦h(x) and h′◦h(1) = 0.In this case (S,T2, ν) is also a measure theoretic factor of ([0, 1],T1, ν ◦ h−1) forsome T1 invariant Borel measure ν. Let us now assume that µ(T−1

1 (1)) = µ({1}) = 0for a T1 invariant measure µ on [0, 1] and define Y1 ≔ [0, 1]\

⋃n∈N T−n

1 (1). ThenT−1

1 (Y1) ⊆ [0, 1]\⋃

n∈N T−n1 (1) = Y1 and as 1 ∉ Y1, we have h′ ◦ h = Id, moreover

T2(h(Y1)) = h ◦ T1 ◦ h′−1 ◦ h′(Y1) ⊆ h(Y1). Therefore the systems (Y1,T1, µ) and(Y2,T2, µ ◦ h−1) have a conjugacy map h|Y1 and especially h|Y1 ◦ T1 = T2 ◦ h|Y1 ,which let us conclude that ([0, 1],T1, µ) is measure theoretically isomorphic to(S1,T2, µ ◦ h−1).

If (X2,B2,T2, µ2) is (measure theoretically) isomorphic to (X1,B,T1, µ1) andone of the systems is ergodic, weakly mixing, or strongly mixing respectively,then the other fulfils the same property by a similar calculation to (A.1).

Appendix B

Functional analysis

B.1 Separability of Lp spacesLet (X,B, µ) be a measure space, where we assume a second countable topologywith base O = {Un : n ∈ N} and µ being regular. By the assumptions for everyA ∈ B is µ(A) = inf{µ(

⋃i∈I Uni) : A ⊆

⋃i∈I Uni , I ⊆ N}. By [31, Satz 2.28a)]

the span of {1A : A ∈ B, µ(A) < ∞} is dense in Lpµ(X) for all 0 < p < ∞.

Hence for all f ∈ Lpµ(X), ε > 0 exists g =

∑mk=1 ck1Ai , ck ∈ C, Ak ∈ B, 0 ≤

k ≤ m such that ∥g − f ∥p ≤ ε. For all of these Ak, 0 ≤ k ≤ m exists an Ik ⊆ Nwith |µ(A) − µ(

⋃i∈Ik

Uni)| < ε/(m|ck|). Hence by defining g′ ≔∑m

k=0 ck1⋃

i∈IkUni

,∥g − g′∥p <

∑mk=0 |ck|ε/(m|ck|) = ε. An approximation of f by a linear combination

stemming from the base O is now possible by ∥ f − g′∥p = ∥ f − g + g − g′∥p ≤ 2ε.As a result the span of all U ∈ O with finite measure is dense in Lp

µ(X) and we gethold onto the following theorem.

Theorem B.1.1. Let (X,B, µ) be a measure space with second countable topologyand µ being regular. Then Lp

µ(X) is separable for all 1 ≤ p < ∞.

B.2 Topologies and metricsEach normed space is a metric space and each metric space is a topological space,by defining a base for the topology via open balls given by the metric. Each topo-logical space induces a measurable space, which is further pursued in Chapter 4.Let X be a locally compact Hausdorff space and K ∈ {C,R}. Once and for allthe spaces Cb(X,K) of bounded continuous functions and Cc(X,K) of continuousfunctions with compact support are equipped with ∥ · ∥∞ and they are often writ-ten without their image space in notion. As Cc(X) ⊆ Cb(X) it carries the inducedtopology. The closure of Cc(X) in Cb(X) is given by C0(X) = { f ∈ Cb(X) : ∀ε >

171

172 APPENDIX B. FUNCTIONAL ANALYSIS

0∃g ∈ Cc(X) with ∥ f − g∥∞ ≤ ε}. Heuristically speaking, C0(X) consists of allcontinuous functions that vanish at infinity. On the space C(X) of all continuousfunctions on X a subbase, for what is called the compact open topology, is givenvia { f ∈ C(X) : f (K) ⊆ U}, K ⊆ X compact, U ⊆ K open. The compact opentopology is equivalent to the topology of compact convergence, that is uniformconvergence on compact sets, see [49, Ch. 7, Thm. 11]; and equivalent to theinitial topology given by the inclusions of C(X :K) ≔ { f ∈ Cc(X) : supp( f ) ⊆ K}into C(X), see [49, Ch. 6, Thm. 10]. The spaces C(X :K) are equipped with ∥ · ∥∞and K ⊆ X always denotes a compact set. It is therefore the natural understandingof local uniform convergence on C(X).

B.2.1 Normed spacesFor normed vector spaces B a topology is induced by their norm and, throughoutthis work, this will always be the assumed topology, unless explicitly stated oth-erwise. For two normed spaces (B1, ∥ · ∥B1), (B2, ∥ · ∥B2) define L (B1, B2) as thespace of all continuous linear operators form B1 to B2 with respect to ∥T∥L ≔∥T∥L (B1,B2) ≔ sup∥x∥B1≤1 ∥T (x)∥B2 for all T ∈ L (B1, B2), see also Section 6.1. Withthat the tuple (L (B1, B2), ∥ · ∥L ) is a normed space, the induced topology is calledthe strong topology and is complete if B2 is complete, [86, Satz II.1.4]. In caseB2 is R or C the space L (B1, B2) is always a Banach space, its elements T areusually called (linear) functionals and we define for any η ∈ L (B1,K) and f ∈ B1

the dual paring ⟨η, f ⟩ ≔ η( f ). In particular L (C0(X,C),C), the space consistingof all complex valued continuous linear functionals on C0(X,C) is isomorphic toM (X) the space consisting of all complex valued regular Borel measures on X,see Theorem 4.3.2.

Remark B.2.1. For any T ∈ L (Lpµ(G)) T , by linearity

∥T ( f )∥p

∥ f ∥p=λ∥T ( f )∥p

λ∥ f ∥p=∥T (λ f )∥p

∥λ f ∥p,

for any λ ∈ C and hence

∥T∥L = sup{∥T ( f )∥p : f ∈ Lpµ(G), ∥ f ∥p ≤ 1}

= sup{∥T ( f )∥p : f ∈ Lpµ(G), ∥ f ∥p = 1}

= sup{∥T ( f )∥p/∥ f ∥p : f ∈ Lpµ(G), f ≠ 0}.

B.2.2 Vague topologyWhile ∥ · ∥L induces the strong topology on the space of all linear functionals, wealso rely on the weak topology for convergence of linear functionals η : B → C.

B.3. THE SPACE C′C

173

There, a sequence of linear functionals (ηn)n∈N converges to η if and only if forall f ∈ B the pointwise limit limn→∞⟨ηn, f ⟩ = ⟨η, f ⟩ exists and is referred to asweak-∗-convergence. For X = Cc(X,C) we define C′c(X,C) as the space of alllinear functionals η of which for all compact K ⊆ X exists a constant CK > 0 suchthat |⟨η, f ⟩| ≤ CK∥ f ∥∞ for every f ∈ C(X : K). The space C′c(X,C) is closed withrespect to the weak-∗-topology, which will be referred to as the vague-topologywithin this work, see [27, (13.4.1)]. It induces the vague-topology on M(X) byconstruction via Theorem 4.2.1. The corresponding weak-topology on Cc(X) isgiven by the inductive topology of the spaces C(X : K), hence the open compacttopology induced on Cc(X), see also [59, Ch. 1], [75, Appendix C].

If X = G, a local compact abelian group, as in Appendix C.2 or Section 5.1,the space CP(G) of all continuous positive definite functions is contained in Cb(X),see Definition 5.2.1. It will be equipped with the topology of compact conver-gence, that is uniform convergence on every compact set; note that this is ingeneral not the induced topology with respect to ∥ · ∥∞ on Cb(X). The Bochnertransform from CP(G) to M+(Γ) is a homeomorphism, [13, Thm. 3.13]; indeed∥ f ∥∞ = f (0) = µ(G) = ∥µ∥ for any f ∈ CP(G) with Bochner transform µ. As the0-function is positive definite, the Bochner transform is a homeomorphism on thelinear span SCP(G) of CP(G). The transform for measures given by (5.4), (C.2.1)is a homeomorphism between the spaces of all non-negative positive definite reg-ular Borel measures on G and Γ, respectively, when both spaces are equippedwith the topology of vague convergence, [13, Thm. 4.16], which can, again, beextended to their linear spans.

B.3 The space C′cLet X be a locally compact separable completely metrisable topological space,that is a locally compact polish space. Note that in [81] it is only assumed thatX is a complete regular space, but as we also rely on [27] we take the strongerassumptions. If X is also assumed to be compact then Cc(X) = C0(X) and hencethere is a one-to-one correspondence between continuous linear functionals andcomplex-valued measures by Theorem 4.3.2, see also Appendix B.2. If X is notcompact, then this method can only be applied for non-negative continuous linearfunctionals, see Theorem 4.2.1. However, there is the following theorem.

Theorem B.3.1 ([27, (13.1.9)]). Let (Ui)i∈I be an open cover of X and ηi ∈ C′c(Ui)

such that (ηi)|Ui∩U j = (η j)|Ui∩U j for all i, j ∈ I. Then there is a unique η ∈ C′c(X)such that η|Ui = ηi for all i ∈ I.

Let (Ui)i∈I be an open cover of X. As X is locally compact and separable itis σ-compact and hence there exists a sequence (K j) j∈N such that Ui ⊆ K j ⊆ Ui j .

174 APPENDIX B. THE SPACE C′C

With Theorem 4.2.1 any ηi j ∈ Cc(Ui j)′ restricted onto K j corresponds to a unique

complex-valued Radon measure µ j on K j. If the assumptions of the previoustheorem are satisfied, then (µk)Kk∩Kl = (µl)Kk∩Kl for all k, l ∈ J and if the uniquefunctional η ∈ Cc(X) derived in the theorem is evaluated on a function f ∈ Cc(X),with supp( f ) ⊆ K, then ⟨η, f ⟩ = ⟨µ j, f ⟩ =

∫f dµ j for all j ∈ N such that K ⊆ K j.

We abuse notation in this work by writing instead ⟨η, f ⟩ =∫

f dη and implicitlymean the construction just described. There also exists different ways of the onejust mentioned to derive a more general treatment of complex-valued measures,see e.g. [81, Ch. 7,8] or [19].

Note that for all η ∈ C′c(X), there is a unique non-negative functional in C′c(X),hence a unique µ ∈ M(X) such that for all non-negative f ∈ Cc(X) we have⟨µ, f ⟩ = sup{|⟨η, g⟩| : g ∈ Cc(X), |g| ≤ f } , by ([27, (13.3.2-3)]). We denote µ by |η|and say that a complex-valued-B(X)-measurable function f is η integrable, if andonly if it is in L1

|η|(X). Note that there is a decomposition of η into its non-negativereal and imaginary parts which we will denote by ηi, i ∈ {1, 2, 3, 4} and f is ηintegrable if and only if it is integrable with respect to ηi for i ∈ {1, 2, 3, 4}. Thefollowing term is then well defined∫

f dη =∫

f dη1 −

∫f dη2 + i

∫f dη3 − i

∫f dη4

and is referred to as the integral of f with respect to η. We further note that everyfunction of Cc(X) is η-integrable, see ([27, Ch. 16]). The space of such functionswill be referred to as L1

η(X) and for any p ≥ 1, a complex-valued function f isin Lp

η(X) if | f |p ∈ L1η(X). Within this work p is either 1 or 2 and we note that the

space is a Banach spaces in this case, see [27, (13.11.4)]. We like to emphasisethat η ∈ C′c(X) is in general not a measure on the Borel-σ-algebra on X, as onecan only make sense of η(A) for any relatively compact Borel set A ⊆ X.

Appendix C

Fourier methods

C.1 ConvolutionLet G be a locally compact, Hausdorff, σ-compact abelian group and let mG bea Haar measure on G. For two Borel measurable functions f , g : G → C, theirconvolution

f ∗ g(u) ≔∫

Gf (x)g(u − x) dmG(x), u ∈ G

is not always well-defined. Therefore the next theorem will present some of themost important cases when convolutions of functions are well defined.

Theorem C.1.1. (a) f ∗g(u) = g∗ f (u), whenever∫| f (x)g(u− x)| dmG(u) < ∞.

(b) For f , g ∈ Cc(G) we have f ∗ g ∈ Cc(G), especially supp( f ∗ g) ⊆ supp( f ) +supp(g).

(c) For f ∈ Cc(G), g ∈ C(G) we have f ∗ g ∈ C(G).

(d) For f , g ∈ L1mG

we have f ∗ g ∈ L1mG

, in particular ∥ f ∗ g∥1 ≤ ∥ f ∥1 · ∥g∥1.

(e) For f , g, h ∈ L1mG

we have ( f ∗ g) ∗ h = f ∗ (g ∗ h).

(f) For f ∈ L1mG, g ∈ L∞mG

we have f ∗ g ∈ Cb(G) is uniformly continuous.

(g) For f ∈ LpmG , g ∈ Lq

mG , 1/p+1/q = 1 and 1 < p < ∞, we have f ∗g ∈ C0(G).

Proof. All are in [77, 1.1.6], except (c). Taking u, v ∈ G, then

| f ∗ g(u) − f ∗ g(v)| ≤∫

G| f (x)g(u − x) − f (x)g(v − x)| dmG(x)

175

176 APPENDIX C. FOURIER METHODS

≤(maxx∈G

f (x))∫

supp( f )|g(u − x) − g(v − x)| dmG(x).

As g is continuous, for all ε > 0, u ∈ G, there exists an open set U ∋ 0 suchthat for all v ∈ G with u − v ∈ U it follows |g(u − x) − g(v − x)| < ε. And| f ∗ g(u) − f ∗ g(v)| ≤ ε maxx∈G f (x) mG(supp( f )) completes the proof. �

Definition C.1.2. A functional η ∈ C′c(G) is called translation bounded, if for allcompact K ⊆ G it exists a CK > 0 such that for each f ∈ Cc(G) with supp( f ) ⊆ K

supy∈G|⟨η, f (y + ·)⟩| ≤ CK ∥ f ∥∞,

and y ↦→ ⟨η, f (y + ·)⟩ is a continuous function on G.

This definition is consistent with the definition of translation boundedness in[3] via [3, Thm. 1.1] and shift-bounded in [13, Def. 1.11]. Note that any measureµ ∈ M(G) can be viewed as a functional and is translation bounded if for anycompact K ⊆ G and for all f ∈ Cc(G) with supp( f ) ⊆ K it holds that

supy∈G|⟨µ, f (y + ·)⟩| = sup

y∈G

∫f (y + x) dµ(x)

≤ ∥ f ∥∞ sup

y∈Gµ(y − K) < ∞.

Hence a measure µ ∈ M(G) is also called translation bounded, if for all compactK ⊆ G it exists a CK > 0 such that supx∈G µ(K + x) < CK . Any µ ∈ M (G) istranslation bounded as a functional, as it is finite. With that at hand we can definethe convolution of a translation bounded η ∈ C′c(G) and µ ∈M (G) as a functionalgiven by

⟨µ ∗ η, f ⟩ ≔∫⟨η, f (y + ·)⟩ dµ(y),

for any f ∈ Cc(G). This is well defined, as for all f ∈ Cc(G)

|⟨µ ∗ η, f ⟩| ≤∫|⟨η, f (y + ·)⟩| d|µ|(y) ≤ CK ∥ f ∥∞|µ|(G) < ∞,

where f ∈ C(X:K) for some compact set K ⊆ G. If f mG for some f ∈ Cc(G), then

⟨ f mG ∗ η, g⟩ =∫

g(x + y) f (y) dmG(y) dη(x) =∫

g(y) f (y − x) dmG(y) dη(x),

for all g ∈ Cc(G). In this spirit we define the convolution of a translation boundedfunctional η ∈ C′c(G) and a function f ∈ Cc(G) to be a function given by η∗ f (y) ≔⟨η, f (y − ·)⟩, y ∈ G. Moreover the mapping f ↦→ η ∗ f into Cb(G) is continuous,see [13, Prp. 1.12], while the function η ∗ f itself is uniformly continuous, see

C.1. CONVOLUTION 177

[13, Prp. 1.19]. With that the convolution of µ ∈M (G) and a translation boundedν ∈M (G) ∪M(G) satisfies for any A ∈ B

µ ∗ ν(A) =∫

G

∫G1A(x + y) dµ(x) dν(y).

A profound problem is that in the case of unbounded measures, even the nicestsettings are in general not locally finite and hence not even Borel measures, as

m ∗ m([0, 1]) =∫R

∫R

1[0,1](x + y)dm(x)dm(y) =∫R

m([0, 1])dm = m(R)

suggests. This led to many definitions of convolution for infinite measures usingaveraging techniques with respect to some sequences, such as Følner sequencesand Van-Hove sequences, see e.g. [80, 58, 59]. In this work we will discuss inSection 6.3 convolution averaged along [−N,N] on Z for N → ∞.

C.1.1 Dual group and charactersHere background information to the field of harmonic analysis will be given whichpartly explain the difficulties that arise to generalise the notion of Fourier trans-form. This section uses techniques described in [60, 13, 24]. It provides no in-sights which will proof useful within this work and therefore each reader may feelfree to read it to their own accord. We will start from a general point of view andthen see that, by natural assumptions, we end at the unit circle and integers.

In this whole section let G be a locally compact, Hausdorff, σ-compact abeliangroup and G is further assumed to be separable. A character of G is a continuoushomomorphism χ : G → S1 ⊆ C, that is χ(x + y) = χ(x)χ(y) for all x, y ∈ G. Theset of all characters of G equipped with the compact open topology is denotedby Γ and called the group dual to G with multiplication in S1 and may also bedenoted by ˆG. Γ is again a local compact abelian group and we denote the neutralelement, which is the constant one function, by χe = 1. We note that the dualof the dual-group is thereby well-defined and the canonical injection G → ˆΓ,x ↦→ x∗ is an isometric homeomorphism. Note that if G is compact, respectivelycountable, then Γ is countable, respectively compact.

The space Γ ⊆ L1µ(G) for all µ ∈ M (G), since for all χ ∈ Γ we have |χ| =

∥χ∥∞ = 1. Hence we define the operator F : M (G)→ C(Γ)

F (µ)(χ) ≔∫

Gχ(x) dµ(x),

which is usually understood as the Fourier-Stieltjes transform of µ. Note that theFourier transform of functions on G is also covered, by L1

ω(G) ↪→ M (G), where


ω denotes a Haar-measure. This procedure can be repeated by taking a measureν ∈ M (Γ), F (µ) ∈ L1

ν(Γ), then F (µ)ν ∈ M (Γ). In particular for any x∗ ∈ ˆΓ andany µ ∈M (G)

F (F (µ)ν) (x∗) =∫Γ

x∗(χ) d (F (µ)ν) (χ)

=

∫Γ

x∗(χ)F (µ)(χ) dν(χ)

=

∫Γ

∫G

x∗(χ)χ(x) dµ(x) dν(χ)

In order to apply the Fubini-Tonelli theorem∫ ∫|x∗ χ| d|µ| d|ν| = |µ|(G)|ν|(Γ) has

to be finite and thus ν ∈ M (Γ). Moreover, we assume that ν is a Haar measureon Γ, which in return forces Γ to be compact, see [31, Satz 3.12, Satz 3.15]. Notethat at this point G is already countable as the dual of a compact space. With thisat hand, we have for y∗ ∈ˆΓ and ψ ∈ Γ that

y∗(ψ)∫Γ

y∗(χ) dν(χ) =∫Γ

y∗(χ + ψ) dν(χ) =∫Γ

y∗(χ) dν(χ)

And hence∫Γ

y∗ dν = ν(Γ)1{0∗}(y∗). With these strong assumptions the next stepsare straightforward

F (F (µ)ν) (y∗) =∫

G

∫Γ

y∗(χ)x∗(χ) dν(χ) dµ(x)

=

∫G

∫Γ

(y∗ + x∗) dν dµ(x)

=

∫G1{−y∗}(x∗)ν(Γ) dµ(x) = µ ({−y∗}) ν(Γ).

If, further, ν is normalized and we are abusing notation by not distincting betweenG and ˆΓ, we have shown that F 2(µ)(g) = µ(−g). So µ can be fully recovered(up to a constant) by transforming twice, as changing the sign is an isomorphismand therefore depends on the ∗-map. In more concrete cases it is often takencare of by involving the complex conjugate for one direction of the transform,i.e. χ(x) = χ(−x). This explaines at least partly why the setting G = Z, Γ =[0, 1) is well suited in most cases of Fourier theory. In not so restrictive formats,the Fourier transform is defined on a subset of Cc(G)′, see Appendix C.2. If aη ∈ Cc(G)′ can be transformed twice, η can be recovered from ˆη, [3, Thm. 3.4].For non-negative positive definite Radon measures (possibly infinite) the Fouriertransformation is an isomorphism, [13, Thm. 4.16]. If positive definiteness isconsidered for η ∈ Cc(G)′, the notion of transform can be extended to the span

C.2. FOURIER TRANSFORMATION OF FUNCTIONALS 179

of all positive definite η ∈ Cc(G)′. In [3, p. 29] the authors point out that it isnot known if the spanned space is a proper subset or equal to the set of all twicetransformable measures and, to my knowledge, has not been solved to this day.However, equality holds in the case that G = Z, [3, p. 51], which also includesBochner’s Theorem, Theorem 5.2.2.

C.2 Fourier transformation of functionalsIn this section the transform of continuous linear functionals on the space Cc(G),as described in [3], where G denotes a locally compact abelian group which isσ-compact with respect to its Haar measure and Γ denotes its dual, will be intro-duced. The following definition is inspired by [3].

Definition C.2.1. A η ∈ Cc(G)′ is (Fourier) transformable if and only if thereexists a unique ˆη ∈ C′c(Γ) called transform of η, such that

⟨η, f ∗ ˜f ⟩ = ⟨ˆη, | f ∨|2⟩,for all f ∈ Cc(G). The space of all transformables will be denoted by (C′c(G))T ⊆

C′c(G) and the space of all functionals which are the result of some transform, willbe denoted by (Cc(G)′)∧T ⊆ Cc(Γ)′. The inverse Fourier transform of η ∈ (C′c(G))T

is given by ⟨η∨, g⟩ ≔ ⟨ˆη, g(−·)⟩ for all g ∈ Cc(Γ).

The previous assignment is well-defined by [3, Theorem 2.1] and as K2(G) ≔{ f ∗ g : f , g ∈ Cc(G)} is dense in Cc(G) and furthermore K2(G) is a subset of thespan of all f ∗ ˜f , where f ∈ Cc(G) denoted by SCP(G), see (5.3). Note that in[3] the authors writeMT (G) instead of (Cc(G)′)T and MT (Γ) instead of (Cc(G)′)∧T .Take also note that there is made an implicit integrability assumption on | f ∨|2

with respect to ˆη, but if positive definiteness is involved the transform alwaysexists in the following sense. We note that the definition of positive definitenessfor η ∈ C′c(G) is given in Definition 5.2.5, where Z is replaced by G.

Theorem C.2.2 ([3, Thm. 4.1, 4.2]). If η ∈ Cc(G)′ is positive definite, then it istransformable, that is there exists a unique ˆη ∈ (Cc(Γ))′ such that ⟨η, f ∗ ˜f ⟩ =⟨ˆη, | f ∨|2⟩ for all f ∈ Cc(G). In particular ˆη is positive definite.

The proof uses that the convolution η∗ f ∗ ˜f for f ∈ Cc(G) is a positive definitefunction, see e.g. [13, Prp. 4.4], and then makes use of Bochner’s Theorem toyield the transform. In perspective of the previous theorem we have that any func-tional obtained as a linear combination of positive definite functionals in Cc(G)′

is transformable. The space SCP(G) can be included as a space of densities withrespect to Haar measure into (Cc(G)′)T and in this case the Bochner transformcoincides with the transform we have just defined, [3, Thm. 2.2]. Bochner’s The-orem plays an important role if we want to see if a functional is in (Cc(G)′)∧T .


Lemma C.2.3 (Inverse transform). Let θ ∈ (C′c(Γ))T . One has θ ∈ (C′c(G))∧T if andonly if θ is translation bounded and (y ↦→ ⟨θ,ˆg ∗ δy⟩) ∈ SCP(Γ) for all g ∈ K2(G).In this case there exists a unique η ∈ (Cc(G)′)T with θ∨ = η. In particular we have

⟨θ,ˆg(−·)⟩ = ⟨θ∨, g⟩ = ⟨η, g⟩

where g ∈ K2(G).

Proof. The proof, while it uses the same techniques as the proof of [3, Thm. 2.6],may as well be compared to the proof of [77, Ch.1.5.1, Thm.].

Given the assumptions, by Bochner’s theorem, there exists a unique finite mea-sure ηg ∈M (G) such that θ ∗ˆg(γ) =

∫G

(x, γ) dηg(x). For any f ∈ L1mΓ(Γ),∫

Γ

f (γ)ˆg ∗ θ(γ) dmΓ(γ) =∫Γ

f (γ)∫

G(x, γ) dηg(x) dmΓ(γ)

=

∫G

∫Γ

f (γ)(x, γ) dmΓ(γ) dηg(x) =∫

G

ˆf (x) dηg(x).

This calculation made use of Fubini-Tonelli’s theorem, which conditions are satis-fied, by

∫Γ

∫G| f (γ)(x, γ)| d|ηg|(x) dmΓ(γ) ≤

∫Γ| f | · |ηg|(G) dmΓ = |ηg|(G)

∫Γ| f | dmΓ <

∞, as ηg is finite. This gives for any h ∈ K2(G)

ˆh ∗ˆg ∗ θ(γ) =∫Γ

h∨(y − γ)ˆg ∗ θ(y) dmΓ(y) =∫

G(γ, x)h(x) dηg(x), (C.1)

where we have used the Fourier inversion theorem for functions, [77, Ch.1.5.1,Thm.]. Hence, together with ˆh ∗ (ˆg ∗ θ) = (ˆh ∗ˆg) ∗ θ = (ˆg ∗ˆh) ∗ θ = ˆg ∗ (ˆh ∗ θ)for g, h ∈ K2(Γ), as θ is translation bounded, [3, Thm. 1.2]; one can deduce∫

G(γ, x)h(x) dηg(x) =

∫G

(γ, x)g(x) dηh(x). The last equality holds for all γ ∈ Γ,which let us follow hηg = gηh, as the association in (C.1) is unique by the Fourierinversion theorem. One checks that the equality implies 1{g≠0}1{h≠0}hηg = hηg =

gηh = 1{g≠0}1{h≠0}gηh. And hence the association η ≔ 1{g≠0}1gηg = 1{h≠0}

1hηh is

well-defined and independent of the choice of the function in K2(G). Taking theRadon-Nikodym derivative ∂ηg/∂(gη) = ∂(1/gηg)/∂η = 1, it follows ηg = gη. Thefirst statement then follows from the equation θ ∗ˆg(γ) =

∫G

(x, γ) dηg(x) given atthe beginning, together with γ = 0.

For the second assertion we refer the reader to [3, Thm. 3.4]. There uniqueness(existence) of the Fourier transform of θ is checked and that (ˆη)∨ = η. As the firststatement of the theorem can be written as ⟨η, f ∗ ˜f ⟩ = ⟨θ, | f ∨|2⟩ for all f ∈ Cc(G)it is clear that θ =ˆη and hence ⟨ˆθ, f ⟩ = ⟨η, f (−·)⟩. �

Remark C.2.4. We remark that the space of all Fourier transformable function-als, whose transforms are again Fourier transformable is a subspace of (Cc(G)′)T ∩

(Cc(Γ)′)∧T and contains SCP(G) as densities with respect to Haar measure. Further-more, all these measures are translation bounded [3, Cor. 3.1, Thm. 3.5]. If G isdiscrete then it coincides with SCP(G), [3, p. 51].

C.3. SOME PROPERTIES OF FOURIER TRANSFORMATION 181

C.3 Some properties of Fourier transformationIn this section some basic properties of the Fourier transformation for functionsand functionals are presented, which are used throughout the work.

Proposition C.3.1. Let q, r ∈ Z, then the Bochner transform ˆδqZ+r of δqZ+r existsand is given by ˆδqZ+r =

1qe−r δ 1

qZq, where ey(x) ≔ e2πixy.

Proof. That ˆδqZ+r exists is due to δqZ+r = δqZ ∗δr ∈ SFP(Z). Let ϕ ∈ C([0, 1)), then

⟨δqZ+r,ˆϕ⟩ =∑z∈Z

ˆϕ(qz + r)

=∑z∈Z

∫ 1

0ϕ(y) e−2πi(qz+r)y dy =

∑z∈Z

∫ 1

0ϕ(y) e−2πiry e−2πiqzy dy

=∑z∈Z

q−1∑k=0

∫ k+1q

kq

ϕ(y + k/q) e−2πir(y+ kq ) e−2πiqz(y+ k

q ) dy

=∑z∈Z

q−1∑k=0

∫ k+1q

kq

ϕ(y + k/q) e−2πir(y+ kq ) e−2πiqzy dy

=∑z∈Z

1q

∫ 1

0

q−1∑k=0

ϕ(

w+kq

)e−2πir w+k

q e−2πizw dw

=∑z∈Z

q−1∑k=0

1q

∫ 1

0ψ(w) e−2πizw dw

=1q

∑z∈Z

ˆψ(z) e0 =1qψ(0) =

1q

q−1∑k=0

ϕ(

kq

)e−2πir k

q =

⟨q−1e−r(k/q)δ 1

qZq, ϕ

⟩,

where ψ(w) ≔∑q−1

k=0 ϕ(

w+kq

)e−2πir w+k

q . �

Lemma C.3.2. Suppose η ∈ CFP(Z) and denote by sr : Z→ Z the map x ↦→ x+ rfor all r ∈ Z. Then η◦sr has a Bochner transform given by er ˆη, where er(x) = e2πirx

for all x ∈ [0, 1).

Proof. Let f ∈ CP(Z), then

⟨η ◦ sr, f ⟩ = ⟨η, f ◦ s−r⟩ = ⟨ˆη, ( f ◦ s−r)∨⟩= ⟨ˆη, er f ∨⟩ = ⟨er ˆη, f ∨⟩,

as for all x ∈ [0, 1]

( f ◦ s−r)∨(x) =∫

f (y − r) e2πixy dm(y) =∫

f (y) e2πix(y+r) dm(y) = er(x) f ∨(x).

�


Proposition C.3.3 (Poisson summation formula). Let f ∈ Cc(R,C), then∑z∈Z

ˆf (z) =∑z∈Z

f (z).

Proof. Define g : [0, 1) → C to be g(x) ≔∑

z∈Z f (x + z), which is well-defined,since f has compact support.∑

z∈Z

f (0 + z) =g(0)

=∑z∈Z

ˆg(z)e2πiz0

=∑z∈Z

∫ 1

0g(y)e−2πizy dm(y)

=∑z∈Z

∫ 1

0

∑m∈Z

f (y + m)e−2πiz(y+m) dm(y)

=∑z∈Z

∑m∈Z

∫ m+1

mf (y)e−2πizy dm(y)

=∑z∈Z

∫R

f (y)e−2πizy dm(y) =∑z∈Z

ˆf (z).

Where the dominated convergence theorem holds, as∫ ∑

m∈Z |1[m,m+1)(y) f (y +m)| dm|[0,1)(y) ≤

∫⌈supp( f )⌉ sup | f | dm|[0,1) < ∞, which allows to interchange

summation and integration. �

Danksagung

Ich mochte mich bei Johannes dafur bedanken, dass er mich das erste Jahr uberbegleitet hat. Außerdem danke ich Christina und Lars fur die vielen anregendenDiskussionen. Dies schließt Hendrik mit ein, der mich daruber hinaus besondersin der Abschlussphase meiner Arbeit unterstutzt hat. Auch mochte ich gerne Mo-ritz und Konstantin danken, die ich noch am Ende meiner Dissertation kennenlernen durfte. Großer Dank gilt Kathryn, die mich in allen Fragen rund um dieUniversitat betreut hat. Des Weiteren mochte ich Tony fur viele fruchtbringendeDiskussionen und die Zeit danken, die er mir zu Verfugung gestellt hat. Besonde-rer Dank gilt noch Malte Steffens, der mich die ganze Zeit uber begleitet hat undohne den die Quest sicherlich ganz anders verlaufen ware. Abschließend mochteich mich noch bei Nicolae Strungaru bedanken, den ich wahrend Oberwolfachnaher kennen lernen durfte. Schlussendlich mochte ich noch besonders Marc dan-ken, der mir diesen Pfad erst eroffnet hat und Daniel Lenz fur seine bereitwilligeUnterstutzung.

183

184 NOMENCLATURE

Nomenclature

∆ Parameter space for (intermediate) β-transformationsδC

∑c∈C δc, where C ⊆ X is discrete and at most countable

δc Dirac point mass at c, where c ∈ Xη(n,r)(z) For an η ∈ C′c(Z) given by η(zn) if it exists an zn ∈ Z such that

z = zn + r and 0 otherwise, see Definition 8.4.4γ The T -discontinuity of a β-transformation, γ = dn−k+1 = (1 − α)/βγ, γηy , γT Autocorrelation of the return time comb ηy with respect to T and

reference point y, given by∑

z∈Z Ξ(T, ν)(z) δz, see Theorem 6.3.2γu Autocorrelation of a word u, see Definition 6.6.8κ

p/ql A finite word encoding rational rotation, see Definition 2.3.7µ A Borel measure or functional on Cc

µ(−·) A ↦→ µ(−A) for all A ∈ B(G)ν A Borel measure or functional on Cc

ωjl A finite word associated with a rigid rotation, see Definition 2.3.2

φ(m, j, r, t) Given by ( jqm+1 + qm)tr ∑am+2l= j (lqm+1 + qm)−t, see (3.18)

Φ1 Operator used in the spectral decomposition of the Perron-Frobeniusoperator, see Theorem 6.5.7

φi A function given by ϕm,im ◦ . . . ◦ ϕ0,i0 for m ∈ N, see Theorem 8.4.10Ψ Operator used in the spectral decomposition of the Perron-Frobenius

operator, see Theorem 6.5.7ψ(r) Given by sup{ψw(r) : w ∈ X}, see (3.10)ψw(r) Given by lim supv→w dξ,t(w, v)/dt(w, v)r, see (3.10)ψx,n(r) Given by dξ,t(x, S |Ln |(y))/dt(x, S |Ln |(y))r, see Definition 3.3.7

ψ( j)x,n(r) Given by dξ,t(x, S (a2(n+1)− j+1)|Ln+1 |(y))/dt(x, S (a2(n+1)− j+1)|Ln+1 |(y))r; (3.16)

ψy,n(r) Given by dξ,t(S |Rn |(x), y)/dt(S |Rn |(x), y)r, see Definition 3.3.7

185

186 NOMENCLATURE

ψ(i)y,n(r) Given by dξ,t(S (a2(n+1)−1−i+1)|Rn |(x), y)/dt(S (a2(n+1)−1−i+1)|Rn |(x), y)r; (3.16)ρ Substitution which maps 0 ↦→ 01, 1 ↦→ 1ηy Return time measure, see Definition 6.3.1Σ A finite set, often called alphabet and its elements are often called

lettersσ A substitution as an element of the set Q

Σ∗ Set of finite words for a finite alphabet ΣσL (T ) The spectrum of and operator T , see Section 6.1.1τ Substitution which maps 0 ↦→ 0, 1 ↦→ 10τTM Substitution which maps 0 ↦→ 01, 1 ↦→ 10θ Substitution which maps 0 ↦→ 1, 1 ↦→ 0Θα The set Θ

α∩ Θα of well-approximable numbers of α-type

ϕi, j The function ιi ◦ T ni−1− ji from Ii, j → [0, 1] where 1 ≤ j ≤ ni, see

Theorem 8.4.10, Lemma 8.3.3ϱ Spectral (return) measure of the Thue-Morse substitution, see Propo-

sition 8.5.16ϱ f Spectral measure with respect to a function f , see Definition 6.2.1ϱα(t) A function on R given by 0, if t ≤ 1 − 1/α, by 1 − (α − 1)/(αt) if

1 − 1/α < t < 1 and by 1/α if t ≥ 1, see Section 3.3.3ϱmax Maximal spectral type of an operator, see Definition 6.4.2Ξ(T, ν) Weight function of an autocorrelation of a return time comb, see Sec-

tion 6.3ζ A primitive substitution

ℓα Number to determine if a subshift is α-repulsive; Definition 2.2.14C The set of complex numbersN {0, 1, 2, . . .}N+ {1, 2, 3, . . .}Q The set of rational numbersR The set of real numbersZ The set of integersZq Group with one generator of period qC′c(X) The space of continuous linear functionals with respect to the vague-

topology, see Appendix B.2.2

NOMENCLATURE 187

C(X) Space of continuous functions from X to CCb(X) Space of bounded continuous functions from X to CCc(X) Space of continuous functions from X to C with compact supportJcβ Jψ for ψ(y) = cy−β, where β > 2 and c > 0, see Definition 3.4.1Jψ The ψ-Jarnık set, see Definition 3.4.1L(u) The set containing all factors of uL(X) The language of the subshift X, see Definition 2.2.7Lk(ξ),Lk Approximation of y in the subshift of slope ξ by substitutions with

the first 2k − 1 continued fraction entries, see Definition 3.2.1Rk(ξ),Rk Approximation of x in the subshift of slope ξ by substitutions with

the first 2k continued fraction entries, see Definition 3.2.1T Topology of a spaceh(ζ) The height of a primitive substitution ζ of constant length q; Defini-

tion 6.6.10M(X) The space of non-negative Borel measures on XM1(X) Space of Borel-probability measures on XB,B(X) Borel σ-algebra generated by the topology T , denoted by σ(T )BN The tail-σ-algebra of T given by

⋂n∈N T−n(B), see Chapter 6

L (X,Y) Space of continuous linear operators, see Section 6.1 and Appendix B.2.1L (B) Space of continuous linear operators on a Banach algebra B to B, see

Section 6.1M (X) The space of complex-valued regular Borel measures on XQ A space of substitutions associated with rational rotations, see Defi-

nition 8.5.2

[w]T The set T−1w ([0, 1)), see Section 8.1

[n]q, [n] An element of Zq w.r.t. the canonical projection Z→ (Z/qZ)△ A △ B ≔ (A\B) ∪ (B\A) where A, B ⊆ X1A Maps x to 1 if x ∈ A and to 0 if x ∉ Aη ∗ f The function y ↦→ ⟨η, f (y − ·)⟩, where f ∈ Cc(G)δc ∗ η For all f ∈ Cc(G) given by ⟨δc ∗ η, f ⟩ = ⟨η, δ−c ∗ f ⟩δc ∗ f A function given by x ↦→ f (x − c)⟨µ, f ⟩ Given by

∫f dµ

⟨µ, f ⟩ Given by∫

f dµ

188 NOMENCLATURE

⟨˜µ, f ⟩ Given by∫ ˜f dµ

⟨ f , g⟩ Given by∫

f g dµ

⟨µ ∗ ν, f ⟩ Given by∫

f (x + y) dµ(x) dν(y), f ∈ Cc

⌈x⌉ ⌈x⌉ = z ∈ Z such that x − z + 1 ∈ [0, 1), where x ∈ R⌊x⌋ ⌊x⌋ = z ∈ Z such that x − z ∈ [0, 1), where x ∈ Rµ ∗ η For all f ∈ Cc(G) given by ⟨µ ∗ η, f ⟩ = ⟨η, µ− ∗ f ⟩, where µ−(A) =

µ(−A) for all A ∈ B

µ ∗ ν Given by µ ∗ ν(A) ≔∫1A(x + y) dµ(x) dν(y), where A ∈ B

µ ∗ f Given by µ ∗ f (y) ≔∫

f (y − x) dµ(x), where y ∈ G

Θα The set {ξ ∈ [0, 1] : Aα(ξ) < ∞} of well-approximable numbers ofα-type

A Closure of the set Af (x) Given by the mapping x ↦→ f (x), the complex conjugate of f (x)Θα

The set {ξ ∈ [0, 1] : 0 < Aα(ξ)} of well-approximable numbers ofα-typeˆγ,ˆγηy , ˆγT Spectral return comb or Bochner transform of the autocorrelationγ, γηy and γT respectively, see Definition 6.3.3˜f (x) Given by the mapping x ↦→ f (−x)

{x} The fractional part of x{x} The set containing the point xA◦ Interior of the set Af ∗ g Convolution of functions, f ∗ g(y) ≔

∫f (x) · g(y − x) dmG(x)

v ∧ w The longest common prefix of v and w

BV The space of functions of bounded variation, see Proposition 6.5.1Aα(ξ) For a subshift of slope ξ given by lim supn→∞ anq1−α

n−1

CP(G) Continuous positive definite functions on GdimH Hausdorff dimensionExact(β) Given by J1

β \⋃

n≥2, n∈N+ Jn/(n+1)β , see Definition 3.4.1

P(G) Positive definite functions on GResL (T ) The resolvent of an operator T , see Section 6.1.1SCP(G) Span of continuous positive definite functions on GSFP(G) Span of positive definite functionals on G

NOMENCLATURE 189

var( f ) Variation of a function f , see Section 6.5A+ {x ∈ A : x is non-negative}, for some vector space V ⊇ AAα,n See Definition 2.2.14Cn A certain subset of ∆, see Definition 8.2.1Dℓ A certain subset of ∆ for a finite sequence ℓ ≔

((ki, ni)

)mi=0, see Defi-

nition 8.4.7di Discontinuities of T n, where T (x) = {βx + α}, (β, α) ∈ Cn and 1 ≤

i ≤ n + 1, see Definition 8.2.2dξ,t(v,w) The spectral metric of v and w given by |v ∨ w|−t +

∑n>|v∨w| bn(v)n−t +∑

n>|v∨w| bn(w)n−t, see Definition 3.3.2Dk,n Given by {(β, α) ∈ Cn : zk, zk+1 ∉ (T (0),T (1))}, see Definition 8.2.8ey For any x ∈ R given by e2πixy, domain is often [0, 1)f Usually a function of some space into C or Rf (−·) x ↦→ f (−x) for all x ∈ Gfv(u) Frequency of v in u, see Definition 2.2.2G Locally compact, Hausdorff, σ-compact abelian groupg Usually a function of some space into C or Rh Normalised eigenfunction of the Perron-Frobenius operator, given in

Theorem 6.5.7h The Parry density for a dynamical system of a β-transformation, see

(8.2)h Topological pressureIi Given by [T i(0),T i(1)) for i ∈ {1, . . . , n−1} and [0,T n(1))∪[T n(0), 1)

for i = n, see Lemma 8.3.3m Lebesgue measure on R or [0, 1) ≅ R/ZmG A fixed Haar measure on GP The Perron-Frobenius operator, see Equation (6.12)pn, qn Defined for a continued fraction such that pn/qn = [0; a1, . . . , an]pu, p The complexity function of a word u, see Definition 2.2.4q Often the denominator or length for a substition of constant lengthQ(n) See Definition 2.2.14Qα Number to determine if a subshift is α-finite, see Definition 2.2.14R Repetitive function for some subshift, see Definition 2.2.10Rα Number to determine if a subshift is α-repetitive; Definition 2.2.12S The left shift operator on ΣN or σ∗

190 NOMENCLATURE

sb,a For any x ∈ R given by bx + a, domain is often [0, 1)sb For any x ∈ R given by bx, hence sb = sb,0

T A map that is used for the dynamics of a system. That includestransformations, β-transformations and operators respectively

T,Tβ,α The mapping x ↦→ {βx + α} on [0, 1) or [0, 1], with 1 ↦→ β − 1 + αTα The map x ↦→ {x + α}, α > 0Tµ Transfer operator of a measure µ and a transformation T , see Sec-

tion 6.5UT Koopman operator of a transformation T , see Section 6.5X A locally compact separable completely metrisable topological spaceX A subshift, that is a shift-invariant closed subset of ΣN

x Given by limk→∞ Rk taken with respect to dt, see Definition 3.2.1X′ Space of continuous functionals, see Appendix B.2.1Xζ Subshift of a primitive substitution ζXu Subshift of a word u, usually minimaly Given by limk→∞Lk taken with respect to dt, see Definition 3.2.1y The reference point of a return time comb, see Definition 6.3.1zi For i ∈ {1, . . . , n} the fixed points of T n, where T (x) = {βx + α} and

(β, α) ∈ Cn, see Definition 8.2.5

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